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COPYRIGHT DEPOSIT 



A PRIMER 



OF 



CALCULUS 



BY 



ARTHUR S. HATHAWAY 

PROFESSOR OF MATHEMATICS IN THE ROSE POLYTECHNIC 
INSTITUTE, TERRE HAUTE, IND. 



- ' 



^MACMILLAN ±3Hr CO. 

LONDON: MACMILLAN & CO., Ltd. 
1901. 

All rights reserved. 



THE UBtfARYiDlP 
CONGRESS, 

^"w CwHtee RtO»»et 

WAV gg faiH9 

OnBMOWHT WTWr 

o^aj . / S~~ I <[ o l 

|CL/iR# <^XXo No. 

corv b. 



Copyright, 1901, 
By ARTHUR S. HATHAWAY. 



Moore & Langen Printing Co. 
terre haute, ind. 






PREFACE. 



This Primer has been written to meet the needs of 
the author, first, for a primary course in the calculus, 
and secondly, for an outline of topics in a more advanced 
course that is suitable for combined lecture and text 
book instruction. 

The author's method of development is essentially 
Newton's method of fluxions, as presented by Hamilton 
in his Elements of Quaternions, Bk. Ill, ch. II. This 
method is clear, logical, and scientific, and it deserves 
more recognition than it has received in general analysis, 
if for no other reason than that it is the method of the 
original discoverer of the calculus. Its failure to be 
adopted is due to want of early publication and defective 
notation, since it is remarkably perfect and general in 
principle. The subsequent discoverer, Leibintz, gained 
the field by publications in a desirable notation, 
although founded upon inferior infinitessimal principles. 
Lagrange attempted a modification of the infinitessimal 
into the idea of a principal part as determined by first 
terms of expansions, and made the " differential co- 
efficient" the primary quantity. Modern text books 
have returned to Newton's method of limits as applied 
to Lagrange's differential co-efficient; there is here 
offered a complete return to Newton, with the fluxion 
or differential as the primary quantity. 

The point of view of our development is that differ- 
entiation is an arithmetical process and that its resulting 
differentials are numbers like other numbers, which are 
classified as independent or dependent variables ac- 



iv PREFACE 

cording to the like character of the variables from which 
they are derived. The usefulness of a process consists 
in its practical applications, but nothing is gained by 
attempting to show practical utility before the funda- 
mental principles and rules of differentiation are fairly 
mastered. It must be accepted at first that differentia- 
tion is of as much higher order of practical value than 
the usual processes of arithmetic as it is in advance of 
those processes in respect to fundamental ideas. Also, 
the student will have more confidence in the use of the 
calculus when he learns it first as a rigorous and exact 
arithmetical method. The following course of lessons 
brings the student as rapidly as it is desirable into 
practical applications. 

One object of the author has been to discourage 
empirical acquisition through illustrated examples 
worked out in full. If a student is not able to follow 
out careful instructions as to how to do his work, 
without having it done for him, he is lacking in the 
first elements of an engineer at least, and it is time that 
he began mental training in that direction (even if it is 
a little hard,) 

Attention is called to the note on page 6 which 
explains a general principle of notation based on 
dx 2 =dx . dx. While unusual in the case of trigonom- 
etric functions, yet it is clear; i.e., there is no conflict 
between sin a; 2 = since, sin x and sin .£ 2 = sin (x 2 ). It 
also removes several inconsistencies in trigonometric 
notation that many students do not understand. The 
ordinary notation may be used, however, if desired. 
ARTHUR S. HATHAWAY. 



PRIMARY COURSE 



Lesson 


Chapter 


Articles 


Examples 


1 


I 


1-10 


1 


2 




11-16 


2-4 


3 




17-19 


5-10 


4 




20-23 


11-15 


5 




24 

REVIEW 


17-22 


6 


II 


25-35 


1-6 


7 




30-38 


7-12 


8 




39-43 




9 




39-43 


36-41, 30-32 


10 




44-46 




11 




44-16 

REVIEW 


42-59 


12 




47-51 


1-7 


13 




52, 59 23, 


30-33, 36-39 


14 




60 


43-56 


15 




53-55, 61 13-17 


, 40-42, 48, 49 


16 




62 

REVIEW 


1-59 


17 




65 


65-73 


18 


III 


69-75 


1,2 


19 




76-79 


3-15 


20 






17-21 


21 




80-82 


22-25 



p 19 



p 42 



p 42 



p 79 



vi . PRIMAR Y CO URSE 



22 IV 109-112 1 pll4 

23 

24 

25 

26 

27 

28 



29 V 125-130 1-7 p 124 

30 

31 136-139 1-10 pl34 

For collected rules of differentiation and anti-differen- 
tiation, see pages 138, 139. 



REVIEW 




109-112 


1 


113-114 


2-3 


115-119 


4-5 


120-122 


6-10 


123-124 


11-13 




14-17 




18-19 


REVIEW 




125-130 


1-7 


131-135 




136-139 


1-10 



CONTENTS 



CHAPTER I — Differentiation 

ARTICLES PAGE 

Preliminary Concepts and Definitions . 1-24 1 

Examples 19 

CHAPTER II — Principles and Rules 

Principles 25-35 23 

Rules 36-46 29 

Anti-Differentiation 47-62 34 

Examples . .42 

Reduction Formulas 63 46 

Indeterminate Forms 64 49 

Inverse Principle 1 .... 65 51 

Expansions in Series ..... 66 54 

Maximum, Minimum 67 56 

Remainder in Maclauren's Theorem . 68 60 

CHAPTER III — Concrete Representation 

Functions Connected with a Variable Point of 

a Plane Curve and their Differentials . 69-79 63 

State of Change of a Function . . . 80-82 74 

Curvature 83-84 76 

Differentiation of Directed Quantities . 85-86 78 

Examples 79 

Curve Tracing 87-98 87 

Envelopes 99-108 96 



viii CONTENTS 

CHAPTER IV — Integration 

Summation . . . . . . . 109-113 103 

Integration 114-123 106 

Potential 124 113 

Examples 114 



CHAPTER V 

Successive Differentiation 
Partial Differentiation 

Examples . 
Successive Integration 

Examples . 
Rules of Differentiation 
Rules of Integration . 



125-127 
128-130 



131-140 



119 
121 
124 
126 
134 
138 
139 



A PRIMER OF CALCULUS 



CHAPTER I 
Differentiation 

1. Variable and Constant Quantities. Letters 
that do not denote special numbers, as do the letters 
tt=3.14159. . . and e=2.71828. . ., but which stand for 
undetermined numerical values in a given problem, are 
called variables or constants, according as their values 
are considered to change in that problem or not. Unless 
the contrary is stated, first letters of the alphabet will 
denote constants, as a, b, c, ... , and final letters of the 
alphabet will denote variables, as u, v, iv, x, y, z. 

2. Independent and Dependent Variables. The 
independent variables are those whose values are assigned 
at will, each without reference to the value of any other 
variable. The dependent variables are those whose 
values depend upon and are determined by the values 
of one or more of the independent variables. Thus, 
x, y being independent variables, then u=x 2 ,v=y 2 ,w=xy 
are dependent variables. Every problem in which vari- 
ation is possible has a certain number of independent 
variables ; the remaining variables are dependent, and 
in a general sense, each is expressible in terms of the 
independent variables and constants, so that always as 
many equations connect all the variables as there are 
dependent variables. 

3. Functions. A variable whose value depends upon 
and is determined without ambiguity by the values of 



2 A PRIMER OF CALCULUS 

certain other variables is called a function of those varia- 
bles. Thus x 2 is a function of x, and y 2 is a function of 
y, and xy is a function of x and 2/. In general, any 
expression that involves several variables, whose value 
is computable by means of the values of its component 
variables, is a function of those variables. Conversely, 
any function of given variables can have, for its repre- 
sentation, an expression involving the variables upon 
which it depends. Suppose, for example, a function 
were known for which no expression in terms of its 
variables existed ; then it would be proper to make an 
expression that should always stand for the value of the 
function corresponding to the values of the variables 
given in the expression. This was the case, for example, 
with the logarithmic and trigonometric functions when 
they were first considered, and the special symbols log, 
sin, cos, tan, sec, etc. , have been introduced as charac- 
teristic symbols for these functions, so that sin x denotes 
the value of the function whose characteristic symbol is 
sin, corresponding to any value of x. 

4r. We shall often use letters as characteristic sym- 
bols of undetermined functions, and not as undetermined 
numbers, particularly the letters /, F, because each is 
the first letter in the word "function." Thus fa; will 
denote an undetermined function of x that can be chosen 
as we please, a,ndf(x, y) and undetermined function of x 
and y. To make fx one function or another is to identify 
f with the process or characteristic of the function. 
Thus /x=sin x makes /=sin; fx—x 2 makes /=" square 
of"; fx=x 2 — 2x-)-3, makes /=" square of, minus the 
double of, plus 3"; f(x, y)=3x 2 -{-2xy — y 2 makes f 



DIFFERENTIATION 3 

stand for computing three limes the square of the first 
variable named, plus twice the product of the first and 
second, minus the square of the second. 

5. Original Values. The original values of the 
variables are those which we suppose designated by the 
symbols of the variables, so that they are either unde- 
termined values, or else the numerical values that may 
be assigned to such symbols; thus x, y, x 2 , y 2 , xy, are 
undetermined original values, and x=S, y=4, x 2 =9, 
2/ 2 =16, xy=l2 are assigned original values. The ad- 
vantage of leaving the original values of the variables 
undetermined or literal, when carrying out processes of 
computation with them, is that one such literal devel- 
opment involves the results of all possible assumptions 
of numerical value. 

6. New Values. New values of the variables will 
be new symbols of the variables (in general the old 
symbols accented) that stand for undetermined or de- 
termined values that are in general different from the 
original values. Thus, if x, y be original values of 
two independent variables, then x', y' would denote new 
values of those variables, and x 2 ,y 2 , xy are the original 
values, and x' 2 , y' 2 , x'y' are the new values of the 
squares and the product of the independent variables. 
As numerical cases, taking a;=3, 2/=4, we could then 
take x'=4:, y'=7, so that 3, 4, 9, 16, 12 are original 
values of the independent variables and their squares 
and product, and 4, 7, 16, 49, 28 are new values of the 
same. Again instead of these new values V=4, y'=~, 
take new values x'=3.1, 2/'=4.3, that are nearer the 
original values x=3 y=4, and the corresponding new 



4 A PRIMER OF CALCULUS 

values of their squares and product are 9.61, 18.49, 
13.33, which are also nearer the original values than at 
first. When we speak of the values of the variables, 
without qualifying them as new values, we always mean 
the original values. The original values of the variables 
denote, in other words, the values or the variables that 
are being considered. The new values are temporary 
values that are to be considered as approaching the orig- 
inal values ; and they are introduced, and made nearer 
and nearer the original values, for the purpose of deter- 
mining some questions of variation of the variables at 
their original values, just as, in order to determine the 
motion of a train at a given instant, it is practically 
necessary to consider its motion for a very small time 
thereafter, with the knowledge that greater and greater 
accuracy is attained the smaller this time is taken, so 
long as it can be accurately measured together with the 
corresponding distance passed over by the train. 

7. Differences. The changes of value of variable 
quantities from original to new values are called differences 
of the variables. A difference is denoted by prefixing 
the Greek letter delta (A) to the symbol or literal value 
of the variable. Thus 

A#="the difference of x"=x' — x, 

A?/= u the difference of 2/" =2/' — V- 

A(x2)=" the difference of x 2 "=x' 2 — x 2 . 

A(2/ 2 )="the difference of y 2 "=y'*—y*. 

A(xy)— U the difference of xy ' '=x'y f — xy. 



DIFFERENTIA TION 5 

In the numerical cases 

x=3, y=4, and x'=4, y f —7, 
we have, 

As=l, Ay=S, A(x 2 )=7, A(2/2) =3 3, A(a;y)=16; 

and for the new values x'=3.1, y'=4.S, that are nearer 
the original values, we have the smaller differences, 
Ax=.l, A2/=.3, A(x 2 )=.61,A(v 2 )=2.49, A(a#)=1.33. 

8. T%e difference of an Independent Variable is a New 
Independent Variable. In other words, if x be an inde- 
pendent variable, then Ax=x' — x, may be any change of 
value we please. In fact the new value, x', depends upon 
the original value x, and the change of value Ax, viz., 
x'=x-f-Ax= original value plus the difference or change 
of value. The value of Ax depends neither upon the 
value of x nor upon that of any other variable, but can 
be taken whatever value we please. If, however, x be 
what is called a real independent variable, i. e. , one 
limited to real values only, then Ax must also be a real 
independent variable. In fact, generally, the limitation 
of all values of a variable to real values also limits its 
changes of value or differences to real values. 

9. The difference of a dependent variable is a new de- 
pendent variable, whose independent variables are the origi- 
nal independent variables and their differences. Thus take 
the square of an independent variable, as x 2 , then 
when x and Ax are assigned we have 

x'=x+Ax, and x' 2 =x 2 +2xAx+Ax2 ) * 
so that A(x 2 )=x' 2 — x 2 =2xAx+Ax 2 , 
which depends upon x, A x, as required. 



6 A PRIMER OF CALCULUS 

Similarly A {xy ) = x'y' — xy=(x-\- Ax) (y -J- Ay) — xy 
=xAy-\-yAx-\-AxAy, which depends upon x, y, Ax, Ay, 
as required. In general, if w=f (x, y), then w'=f(x f , y') 

Am?=/ (z+Ax, y+Ay)—f(x, y), 

which depends upon x, y, Ax, Ay. It appears also 
from this result, that even when the variables of a 
function are not independent variables, the difference 
of such function will depend upon its variables and 
their differences in exactly the same way as if the vari- 
ables were independent. 

10. Proportional Differences. Equimultiples of 
simultaneous differences by the same real proportional 
factor will be called proportional differences; they are 

*Such symbols as Ax,fx, sin^, log x, etc., which are not 
separable into number factors, because one of the factor 
symbols is a characteristic, and not a number, are equiva- 
lent to single symbols of number, and exponents to such 
a symbol should be regarded as applying to the symbol as a 
whole, when no parenthesis or dot intervenes to make a 
separation of its parts. Thus A.r 2 =A.rA:r, and not the differ^, 
ence of x % . The latter difference is written A(x 2 ) or A.x 2 . A 
similar symbol is A 2 ^=A Ax= difference of Ax, regarded as a 
new independent or dependent variable according as x is 
independent or dependent. However, by force of usage, 
and contrary to principles of notation, sin 2 x means sinx 2 and 
not sin (sin #), and similarly for the squares or other powers 
of all trigonometric functions, except for the exponent — 1. 
Thus sin -1 :r is not sin of 1 , but conforms again to general 
principals of notation in which / 1 x stands for " that func- 
tion whose f is a:," so that always ff 1 x=x. 



DIFFERENTIATION 7 

any proportionals to the differences, if we understand 
the term j:>roportion in the sense that the ratio is a real 
and not an imaginary number. The literal symbol of the 
proportional factor will be N, so that Nkx, N&y, 
A T A(a; 2 ), JVA(2/ 2 ), NA(xy) denote proportional differ- 
ences of the variables x, y, x 2 , y 2 , xy. 

E. g., let x=3, y=A, Az=l, Ay=3, A=4, then 

NAx, N&y, NA(x 2 ), NA(y 2 ), N&[xy) 
=4, 12, 28, ' 132, 64. 

For the new values x\ y f y =S.l, 4.3, corresponding to 
Ax=.l A?/=.3, which are nearer the original values 
than before, and the larger factor A =49, we ,find, 

N&x, NAy, NA(x 2 ), i\ 7 A(2/2), N&(xy) 
=4.9, 14.7, 29.89, 122.01, 65.17. 

Note that although the differences have all been de- 
creased in value from their first values, yet the corres- 
ponding increase in the proportional factor has left the 
proportional differences about the same as before. 

11. Differentiation. Differentiation is the process 
of finding limits of proportional differences of variable 
quantities, as the differences tend toward zero and the 
proportional factor tends towards infinity. Such limits 
are called differentials of the variables. A differential is 
denoted by prefixing the letter d as characteristic of 
differentiation to the literal value of the variable. Thus, 
for independent variables, x, .y, 

dx=differential of £=lim A r Ax^lim N(x' — x), 
dy=differential of y=lim NAy=lim N(y' — y). 

The hypotheses of this differentiation of independent 



8 A PRIMER OF CALCULUS 

variables, x, y, are firstly, that the differences approach 
zero (or the new values approach the old) while the 
proportional factor correspondingly approaches infinity, 
so that the proportional differences approach limits, and 
secondly, that these limits are designated by dx, dy. 

Turning to dependent variables, we have similarly 

d(x*)=lim iVA(£ 2 )=lim N(pJ* —a; 2 ) 

=lim.[(a;'+aO N(x'— x)~]=2xdx ;* 

%a)=HmiVA(2/2)==lim N(tf* —y 2 ) 

=lim [(t/'+ y)N{y'—yy] =2 y dy) 

d(xy)=\\m NA(a^/)=lim N(x'y'—xy) 

=lim [y'N (x'—x) +xN(y'—y)'] =ydx+xdy. 

As further exercises show similarly that d(x*)=3x 2 dx, 

d\ = -^ djx=^j- x , d(x>yz)=ZxyZdx+Sx*y*dy. 

12. To understand differentiation, and the exact 
signification of the resulting differentials as variable 
numbers, some points in the process of differentiation 
must be discussed more fully, and in particular they 
must be illustrated by numerical values. 

13. The differentials of independent variables are new 
independent variables. In illustration, to make dx=b, we 
may take successively 



^Observe that since x / approaches x, therefore x / -\-x ap- 
proaches 2x, and also that N increases as x / approaches x, 
so that N(x'—x) approaches the limit, dx. The limit of a 
product being the product of the limits of its factors, we 
therefore find that lim [(x'+x).N(x'—x)]=2xdx. 



DIFFERENTIATION 9 

Ax=l, .1, .01, .001, .... limit=0 

iV=4, 49, 499, 4999, .... limit=oo ; 

then NAx=4, 4.9, 4.99; 4.999, .... limit=5. 

Thus, in this way, we determine 
cfa=lim NAx==5. 

If instead of the above series of values of N we should 
take another in which every proportional factor becomes 
double its preceding value so that we have successively, 

JV=8, 98, 998, 9998, 

then with the same series of values of A.r as before wc 
should find, 

NAx=S, 9.8, 9.98, 9.998, .... limit=10, 

which gives another value dx,=lim NAx,=lO, 
which is also double the preceding value of dx. In 
general dx can be determined any value we please with- 
out regard to the value of x or of any other variable, 
since the value of Ax may be assigned at will, and its 
series of values approaching zero, assigned likewise 
as we please, in respect to value and law of contin- 
uation, so that whatever series of values approaching 
infinity may have been already assigned to N, we can 
make the proportional difference NAx take a series of 
successive values that will approach any limit we please. 

Take, for example, Ax= _ , which gives, as N in- 
creases indefinitely, a corresponding value -of Ax that is 



10 A PRIMER OF CALCULUS 

approaching zero as required ; then the series of values of 
N Ax will be 

JN r Ax=a i a, a, . . . . 

whose limit approached is a. Or take 

Ax=a 'N'/(N* +5)=a/ (N+JL) which approaches 

zero; then 

NAx=aN*/(N>+5)===a/(l+jL), 

which approaches a. 

There are, in fact, an innumerable number of different 
ways of making each independent difference approach 
zero, and the common proportional factor approach in- 
finity, so that the proportionals of those differences shall 
each approach any assigned value we please. If, how- 
ever, we are considering a real independent variable, x, 
then since N and Ax are real; therefore NAx is real and 
must approach a real value. In words, the differential 
of a real independent variable is a new real independent 
variable. 

14. Understand that in the differential process such 
as Ax=l, .1, 01, .001, . . . . , andiV=4, 499, 49, 4999, 
. . . . , in which the limit of NAx is sought, we do not 
consider Ax as ever actually zero, or N as actually in- 
finity, so that we are not trying to find a value of 
" infinity times zero." In fact, a little common sense 
will show that since neither zero nor infinity are any 
actual values, therefore "infinity times zero" is a 
phrase that is in itself meaningless. Nor can this 



DIFFERENTIATION 11 

phrase be given a definite meaning in accordance with 
the usual acceptance of zero as denoting the nominal 
limit of a value that becomes smaller and smaller 
without limit of smallness, and of infinity as the nom- 
inal (but not-existing) limit of a value that becomes 
larger and larger ivithout limit of largeness. Since one 
factor of a product can become smaller and smaller, 
and the other factor larger and larger, so that the 
product shall approach any value we please, it follows 
that even these limit ideas of zero and infinity cannot 
give determinate significance to infinity times zero. 
Should the student see cause, from these facts, to ob- 
ject to the independent differentials as too indeter- 
minate in value for mathematical consideration, then 
the same objection would be equally valid against any 
independent variables, and against the whole idea of 
variation of value, which must be founded on the 
initial idea of certain indeterminate values, which can 
receive or change value at will, and of other related 
values which depend upon these undetermined or inde- 
pendent values. 

15. The differential of a dependent variable is a new 
dependent variable that is dependent upon and determined by 
(i. e. , a function of) -the independent variables and their 
differentials. This result is a matter of definition and 
as a test of differentiability. For the only way in 
which the limit of the dependent proportional differ- 
ence might be changed in value, ivithout changing the 
values of the independent variables and their differentials, 
would be to take different series of values of the inde- 
pendent proportional differences, but not so as to change 



12 A PRIMER OF CALCULUS 

their assigned limits ; and when such variations of ap- 
proach, alone cause variations in the limit of the de- 
pendent proportional difference, we may consider that 
there is no definite limit or differential, and that the de- 
pendent variable is therefore non-differentiable. The 
test of differentiability is therefore the determination of 
the dependent differential solely in terms of the inde- 
pendent variables and their differentials. 

16. For example, a: 2 is differentiable, because d(x 2 )= 
lim iVA(x 2 )===lim (x'-\-x). N(x r — x)=2xdx, an expres- 
sion that is definitely obtained in whatever way we 
suppose x' to take an indefinitely continued series of 
values that approach the limit x at the same time that 
N takes a corresponding series of larger and larger values 
so that N(x' — x) approaches the limit dx. Similarly 
x y is differentiable, because we find invariably 

lim NA(xy)=lim (y' N Ax-\-xN Ay)=y dx-\-x dy, 

in whatever manner Ax, Ay approach zero and TV ap- 
proaches infinity so that we have lim NAx=dx, 
lim NAy=dy, It will be found that all ' ' continuous' ' 
variables,* for which expressions known to the student 
exist, are differentiable, except that in some expres- 
sions, for certain values of the variables involved, it 

*" Continuous" means varying by small amounts when 
the variables change by small amounts, the dependent 
change approaching zero when the independent changes do 
so. The expression fx=x-\- integer part of x, is not continu- 
ous at integral values of x; viz. when x f increases towards 
2,fx / increases toward 3, but/ 2=4, so that f x / — fx does 
not tend to vanish as x / approaches x=2. 



DIFFERENTIATION 13 

may happen that the value of the differential is ambig- 
uous. This will be shown in the differential expres- 
sion itself, so that it need not be regarded as affecting 
the general differentiable character of the variable in 
question. 

17. The differential of a function of one or more varia- 
bles is the same function of its variables and their differen- 
tials, whether the variables are all independent or one or more 
of them are dependent. 

This is a consequence of the definition of differentia- 
bility which makes a function, w=f(x, y) that is differ- 
entiable have a differential, 

dw=\im iVAw;=lim N [f(x-\rAx, y-\-Ay)—f (x, y)~\, 

that is a definite expression in terms of x, y, and dx= 
lim iVAx, dy =lim NAy; say dw=f'(x, y, dx, dy). 
If such a result holds when x, y are independent vari- 
ables, so that we have arbitrary methods of making 
lim NAx=dx, lim NAy=dy, in which dx, dy, are 
arbitrarily selected values, then it must all the more be 
true when we have only certain dependent methods of 
making lim NAx=dx, lim N&y=dy, where dx, dy are 
dependent values. In other words, the dependent 
methods of approach, and the dependent limits, are 
included among the arbitrary methods of approach and 
the arbitrary limits. Thus 

% 2 )= lim (y'+y)NAy=2ydy, 

whether y is independent, so that lim NAy=dy is also 
independent, or whether y is dependent, so that 
lim NA=dy is also dependent. In the latter case there 



14 A PRIMER OF CALCULUS 

remains the finding of dy in proper terms, from the 
value of y in terms of the independent variables, before 
the differentiation of y 2 can be considered as com- 
pleted. Similarly d(xy)=ydx-\-xdy, whether x, y are 
independent or dependent variables ; and if we have 
actually y=x 2 , so that dy=2xdx, then xy~-x 2 > and 
d(x z )—x 2 dx-\-x .2xdx=3x 2 dx. Although this result 
is obtained indirectly yet it must verify directly. Thus, 

N&(x*)=N(x'Z—xZ)=(x' 2 -\-x'x+x 2 )N(%'— x) 

whose limit [as %' approaches x and N increases so that 
N(x! — x) approaches dx~\ is easily seen to be 3x 2 dx. 
Again, in d(xy)=y dx-\-xdy, we can put y—x 3 so that 
dy=-Zx 2 dx, and making these substitutions for y and 
dy, we find 

d ( x 4 ) =x 3 dx-\-x. Sx 2 do:=4x 3 dx. 

Let the student verify this result directly, and also go 
over the differentiation of the product xy which gives 
the value d(xy)=ydx-\-xdy, and try to find how any 
supposed dependence of y upon x could do more than 
make dy correspondingly dependent upon x and dx 
(assuming, of course, that the given dependence of y 
upon x makes it a differentiate function. ) 

18. The differential of a given function is therefore 
seen to be a fixed rule for differentiating that function, 
even when its variables, instead of being simple inde- 
pendent variables, are any complex functions of other 
variables. Thus from d(u 2 )=2 udu, we have it equally 
true, by replacing u by x 2 -\-y 2 , that 



DIFFERENTIATION 15 

which finally reduces to 

since we will find that 

d(x*+y>)=2(xdx+ydy). 

It is on this account that rules of differentiation become 
important and of wide application, whether expressed 
in terms of one set of letters or another, since it will be 
indifferent what letters are employed to denote the 
variables. In fact the more important rules are best 
memorized in words. 

19. Thus, 

d(xy)=ydx+xdy, 

is in words: The differential of a product of variables 
equals the sum of the products consisting severally of the 
differential of each factor into the remaining factor. This 
rule extends, also, to a product of any number of 
factors, e. g., 

d(x y z)=y z dx-{-x z dy-{-x y d z; etc. 

To prove this, let the product x y z change to x' y z, then 
to x' y' 2, then to x' y z'. This is a succession of partial 
changes of value due to first changing x alone, then y 
alone, then z alone, and the sum of these partial changes 
equals the total change. Thus, 

x'y'z'—x y z=(x'yz—x y z )-\ r {x'y'z—x'yz)-{-{x'y'z , —x'y , z) 

or 

A(xy z)=y zAx-{-x'zAy-\-x'y f Az. 



16 A PRIMER OF CALCULUS 

Multiplying this equation by N and remembering that 

d=lim NA, 
and 

we find, 



limx'=x, limy'—y, etc., 



d(x y z)=y z dx-\-x z dy-\-x y dz. 

20. When the values of the variables of a function are 
assigned, then the value the differential of the function varies 
proportionally with the values of the differentials of its 
variables. For, let x, y, ...., be the assigned values of the 
variables, and w the corresponding value of the function; 
then Aw will be assigned when Ax, Ay,... are assigned, 
and lim NAw is determined when lim iVAx, limNAy,.,. 
are determined. If the latter limits be made x 1 , y ir - 
and the former consequently becomes w x , then to make 
the latter change proportionally to kx x , ky 1 ,..., we have 
only to take new multipliers each k times as large as 
before, with the same values of Ax, Ay... as before, since 

lim kNAx—k lim NAx=k x l ,lim kNAy=k lim NAy=ky 1 , 

etc. But in this method of approach each Aw remains 
the same as before, and the limit of the new propor- 
tional difference is 

lim k NA tv=k lim NA iu=k iv x . 

In other words, if w 1: x 13 y^-- be corresponding values 
of dw, dx, dy,..-, and we change dx, dy,... proportionally 
to new values kx x , ky x ,-> then dw changes in the same 
proportion to the new value kw 1 . 



DIFFERENTIATION 17 

21. Since proportional factors must be real num- 
bers, it follows that the proportional factor k of the 
preceding proposition must be real and not imaginary. 
An important consequence of that proposition is that : 
In the differential of a function of one real variable, the differ- 
ential of the variable appears only as a factor of the result, 

or, 

dfx=f'x dx 

where f'x is a function of x called the differential co- 
efficient of fx as to x, and also, the derivitive of fx as to.r. 
In fact x being assigned, if any two values of dx are in 
the ratio k : 1, (where k must be real because x and 
therefore dx are real variables) then the corresponding- 
values of df x are in the same ratio by Art. 20; thus 
the quotient dfx/dx does not change value when dx 
changes value; and thence this quotient depends on 
the value of x alone, so that it is some function, f'x, 
of x. 

22. The theorem of Art. 21 does not hold for all 
functions, when the variable is not limited to real values. 
Thus if z=x-\-yV — 1 be an imaginary variable whose 
real components are x, y, then 

mod z= J ( x 2 -\-y 2 ) 

is a function of z, whose differential will be, as the 
student may verify by the work in full, 

d mod z=(x dx-\-y dy) /mod z. 

If this differential contain dz=dx-\-,J — l.dy as a factor 
only, so as to be of the form f'z dx -f- J — If'z dy what- 
ever values dx, dy may have, then 



18 A PRIMER OF CALCULUS 

f'z=x/ mod z=y/ (mod zj — 1), or x== — yj — 1 

which is impossible, remembering that x, y are any 
real values. On the contrary 

d.z 2 =2zdz, d.z*=3z*dz, etc. 

23. Analytical Functions. Any differentiable 
function of one variable, whose differential contains the 
differential of its variable only as a factor, is called an 
analytical function. Any function of a real variable is 
(art 21) an analytical function; but for an imaginary 
variable z, mod z is not an analytical function of z, 
while z 2 , 2 3 , etc., are such. 

24. Derivation. Derivation is the process of differ- 
entiation followed by division by the differential of a 
variable. The result of derivation is the derivative of 
the function as to the variable, and must be a function of 
the variable alone if derivation is possible. In other 
words, derivation is a process that is applicable only to 
analytical functions of one variable. Derivation can 
have a definition of its own not depending upon • differ- 
entiation, viz., it is the process of finding the limit of 
the quotient of the difference of the function by the 
difference of the variable as the differences approach 
zero, provided there is a definite limit depending on the 
value of the variable alone, and not at all upon the 
manner of approach of its difference to zero. This fol- 
lows from 

dfx ., NAfx .. Afx 
■4- =hm ' = lim — — . 

dx NAx Ax 



DIFFERENTIATION 19 

Examples. I. 

1. If x=3, y=4:, and we take successively Ax-— 1, 
.7, .07, .007, and so on smaller and smaller, Ay— 2, 
1.3, .13, .013, and so on smaller and smaller, then find 
the corresponding series of values of 

A(x*), AG/*), A(xy). 

2. If in Ex. 1, we also take successively 
N=l, 9, 99, 999, and so on, larger and larger 

show that we thus determine 

dx=7, dy=13, d(x*)=£2, c% 2 )=104, d(xy)=67, 
and verify the last three from their literal values 
d(x*)=Zxdx, d(y*)=2ydy, d(xy)=ydx+xdy. 

3. If in Ex. 2 we double each value of N in its series 
of values, show by full numerical computation, that the 
values of dx, dy and also those of the dependent differ- 
entials are doubled. 

4. Show that, if x=3, y=4:, then however we make 
Ax, Ay approach zero and N approach infinity so that 
NAx, NAy, approach 7, 13, respectively, we shall have 
NA(x*), NA(yt), NA(xy) approaching the limits 42, 
104, 67, respectively. 

[N&(x 2 )=N[(S+Ax) 2 — 9~]=6.KAx+Ax.NAx, etc.] 

5. Prove that d(z*)=2zdz, d(y*)=3y*dy, <%-«)= 
6y 5 dy. Also prove the last equation from the preced- 
ing ones, by putting z=y s . 



A PRIMER OF CALCULUS 




6. Prove that 




, 1 dx 7 1 2dy 7 1 
d~= -, d—- = f-, d — = - 

x x 2 y 2 y z x* 


4dx 



Also verify the last equation by taking y=x 2 in the 
second. 

7. Prove that 

djz=dz/2jz, d,J(x 2 +y 2 )=(xdx+ydy)/ t J(ix 2 +y 2 ) 

Also verify the last equation by putting z=x 2 -\-y 2 in the 
first. Also verify the first from (Jz) 2 =z. 

8. Prove that d J (a 2 +s 2 )=sds / ^(a^+s 2 ). 

9. Prove that d.y%=-\y*dy. 

[A.3/i=2/i— 2/5=(2/' 4 — y*) / (j/l+y'iyt+yl^ etc., or 

let w=y*, then w 3 =y*, whence Sw 2 dw=4y s dy, etc.] 

10. Prove that d(x*— 3^ 2 +6x— 4)=3(£ 2 — 2z+2)cfc, 
and that 

d.(x*— 3»»+6a5— 4) 4 = 

12(x— 3o: 2 +6x— 4)3(x 2 — 2x+2)c?^. 

11. Prove that d(ax+^2/)= a ^+^^2/- Thence show 
that the characteristic d of differentiation is distributive 
over a sum and commutative with a constant factor, just 
as if it were a number multiplier. 

12. Prove that d. -= — . State this as a rule 

# x 2 

for differentiating fractions. 

13. Prove that d.^—=-^(2xdy—Sydx). 



DIFFERENTIATION 21 

14. Prove that d^±^p^ = 0, 
also that d [(x + 5) 2 — x* — lOx] =0. 

15. If m, n be any given positive integers, prove that 

m m m m 

7 - m — i 7 , — — m— - — i , 

d.x n = — x n dx, d.x n == x n dx. 

n n 

State this result as a rule for differentiating powers to 
fixed fractional exponents. 

m 

[Let$=x*> theny n = x m , y' n =x' m , and y' n — y n = 
x fm — x m , which may be written, 

(y>n-l + y> i-tyf. . , +y > y n-2 +y n-1) ^_ y) = 
(/m-l_|_ z 'm-2 x _J_ t _ . _j_ x ' x m-2_^_ x m-l) (#'__£) . 

Multipling this by N and proceeding to the limits, 

lim x'=x, lim y f ==y, lim iV(Y — x)=dx, lim iV (2/ — y)=dy 

we find ny n — 1 dy=mx m — 1 dx, and divided by 7/ n =x m , this 

7 , 7 m dx ,' 

is ndy/y=mdx/x, ordy= — 2/— etc. J 

16. Prove that d.x*yi=^x?y%dx-\-% xAy*dy. 

17. Show that the successive derivatives of 

£5_7 x 4_p4 x 3_9 x 2_^2x— 7, 

are 5x 4 — 28x* + 12z2— 18x+2. 
20x3— 84xH-24x— 18, 
60x»— 168a4-24. 
120z— 168, 
120, 
0. 



22 A PRIMER OF CALCULUS 

18. Show that the successive derivatives of (1+j) 4 
are4(l+x) 3 , 12(1+*) a , 24(l+c), 24, 0. 

19. Take various algebraic expressions and differ- 
entiate them by the full process, and also by rules, 
i. e., by examples already worked. 

20. Expand (l-\-x) 4 by derivation. 

[We know that (l±x)±=A+Bx+Cxz+Dx*+Ex*, 
for all values of x, where A, B, C, D, E, stand for some 
unknown numerical coefficients. Deriving this equa- 
tion we find other identities, 

4(l+x)3=B+2Cx+3Vxz+4Exz 
12(l-\-x)*=2C-\-6Dx+12Ex* 
24(l+c)=6Z)+24£a; 
24=24£ 

Taking x=o in these equations, since they are iden- 
tities and so true for all values of x, we find ^4=1, B=4, 
0=6, D=4, E=l, and, 

(l_}-^)4 == l_|-4 x _)_6x 2 +4x3+a;4 # ] 

21. Expand (l-\-x) n by derivation. 

r/< ■ , „ , . n(n — 1) _ . n(n — 1) (w — 2) Q . 
[(l+x)" = l+nx+ K 2] J x* + -± ^ '-x?+.. 

where 21=2.1, 31=3.2.1=6, 41=4.3.2.1=24, etc.] 

22. Expand x z — Zx 2 -\-2x — 1 in ascending powers of 
x — 4, by derivation. 

[ X 3„3^2_|_2£— 1=23+260— 4)+90— 4)2+(£-4) 3 ]. 



PRINCIPLES AND RULES 23 



CHAPTER II 
Principles and Rules 

25. Pkixciple 1. If two variables are always equal, 
or if they always differ by a constant, then their differentials 
are always equal. 

For, let x, y be original values of two variables and 
x', y' any new values as near as we please to the original 
values, then the conditions are, if the variables are 
always equal, that y=x and y'=x', , or if the second 
always exceeds the first by a constant c, that y=x-\-c, 
y'=x'-\-c. In either case y' — y=x' — x, and therefore 
N(y' — y^)=N(x' — x), and as the new values are made 
to approach the old, while N increases so that either 
member approaches a limit, the other member must 
approach the same limit, i. e., dy=dx. 

26. The proof of the above principle shows under 
what circumstances the differentiation of equals gives 
equals, viz., the equation must remain true when the 
variables change from their original values by any cor- 
responding amounts, however small. This principle 
is therefore not applicable to such an equation as 
x' 2 — 3x-f-2=0, which is true for certain values of 
x (x=l or 2), but which does not remain true, when x 
changes from those values. The equations to which 
the principle applies are of three classes, first, absolute 
identities, such as (x-\-y) 2 =x 2 -\-2xy-\-y 2 ; second, limited 



24 A PRIMER OF CALCULUS 

identities which are equal only for certain ranges of value 
of the variables, such as 1/(1 — x)=l-\-x-{-x 2 -\-x z -\-, 
etc., which is true only when x is smaller than 1; and 
thirdly, equations that practically define one of the 
variables in terms of the others, such as x 2 -\-y 2 =a 2 
which makes y= lk J(a 2 — x 2 ). 

27. An alternative form of Principle 1 is that : 

The differential of a constant quantity is identically zero. 

For a constant can be made a function of any vari- 
ables we please, as 2=x-)-2 — x, l=x/x, etc. ; and as 
such a function, its change of value is zero; likewise 
any proportional change of value is zero, and hence the 
limit of such proportional change, or the required dif- 
ferential, is zero. 

28. Inverse Principle 1. If the differentials of two 
variables are always equal, then the variables are either 
always equal or alivays differ by a constant quantity. 

Two proofs of this will be given later, one geometric, 
and one algebraic. An alternative of this inverse prin- 
ciple is: 

(a). If the differential of a quantity is identically zero, 
then that quantity is a constant. 

29. Principle 2. The characteristic, d, of differentia- 
tion, is distributive over a sum, and commutative with a con- 
stant factor. In symbols, 

d(x-\-y)=dx-\-dy, d.ax=adx. 
The proof will be left as an exercise. It is one of the 



PRINCIPLES AND RULES 25 

first results the student would naturally notice in the 
practice of differentiation, and he would probably state 
it in some such form as, the differential of a sum is the 
sum of the differentials of its terms, and, the differential of 
the product of a constant and a variable is the constant into 
the differential of the variable. It is, however, important 
to consider it in the above form as a symbolic law of the 
characteristic d. The second part is really a conse- 
quence of the first, viz., d.2x=d(x-{-x)=dx-{-dx=2 dx, 
etc. 

30. By the partial differentiation as to x, of a function 
of two or more variables x, y, etc., we mean differen- 
tiation as if x were the only variable, and the others 
were constants. The characteristics of partial differ- 
encing and differentiation as to x will be A x , d x , and as 
usual d x = lini iVA x . Thus 

A. c (x 2 2/ 3 )=x' 2 2/ 3 — x 2 2/ 3 , d x (x*y3)=2xy*dx. 
Similarly, 

A y (x 2 2/ 3 )=x 2 7/' 3 — x 2 2/ 3 , dy(x 2 y*)=3x 2 y*dy. 

31. A partial differential as to x is simply a special 
value of the complete differential corresponding to any 
value of dx, and the values dy=o, dz=o, etc., since 
these are the values of dy, dz, that result by making 
y, z, constants. Thus if 

df (x, y, z)=f(x, dx, y, dy, z, dz), 
then 

dxf(.%, V, 2)=/'Oj dx, y, o, z, o), etc. 

32 Principle 3. The complete differential of a fane- 



26 A PRIMER OF CALCULUS 

tion of several variables equals the sum of its partial differ- 
entials as to each variable. In symbols, 

df(x, y, z) =dif(x, y, z)-\-d y f(x, y, z)+d t f(x, y, z). 

For let the complete change be made first by chang- 
ing x alone, then y alone, then z alone, giving the 
successive partial changes from f(x,y,z) to f(x r ,y,z) 
to f(x',y',z) to ■/ <y, y', z') which are denoted by 
A»/(a;,y,-2), & y f(x',y,z), A z f(x',y',z). The com- 
plete change of value of the function is easily seen to 
be the sum of these successive partial changes of value, 
i. e., 

*'/(&, V,z) =*xf (x, y, z) + A y / (x[, y, z) + A, f(x', y',z). 

Multiplying by N, we find an analagous result for the 
proportional differences, which is precisely the principle 
for differentials we wish to prove, except that the 
original values of the variables in the proportional 
differences are in the second, x' instead of x, and in the 
third, x', y' instead of x, y. However, x\ y\ become 
x, y in the limit, and the proportional differences 
become the differentials, so that if the general differ- 
ential is a continuous function of its variables, x\ y' 
will be replaced by x, y in the differentials. For ex- 
ample in N&yf(x',y,z) in which x f , z are treated as con- 
stants, when y' is very nearly y, this proportional dif- 
ference is by definition, very nearly d y f(x f ,y,z), and 
this will be very nearly d y f(x, y,z) when x' is very near 
x, if the latter differential is a continuous function, 
i.e., if df(x,y,z) is a continuous function (of which 



PRINCIPLES AND RULES 27 

d y f(x,y,z) is a special value obtained by making 
dx=o, dz=o.) Thus 

df(x, y, z)=d x f(x, y, z)+d y f(x, y, z)+d z f(x, y, z), 

assuming, as is always the case in the calculus, that the 
functions considered are continuous, so that limits are 
found by substituting the limits of the variables. 

33. That the limit of fx' as x' approaches x is not 
always the same as fx, may be seen from the example 
fx=x-\- integer part of x) taking x=Z and x' less than 
and approaching 2, fx' approaches, 3, but fx=4. This 
cannot occur when fx is continuous, since then by 
definition, fx'—fx approaches zero when x' approaches 
x, and therefore fx' approaches fx. 

34. When x, y, z, are real variables, or, more gener- 
ally, when w=f(x, y, z) is an analytical function of its 
variables whether they are real or imaginary, then by 
Article 21 d x w/dx is independent of dx, and therefore 
a function of x, y, z alone. This quotient is called the 
partial derivative of w as to x, and is denoted by 9w/9x, 
the script d being notice of partial differentation, while 
the denominator shows the variable of differentiation. 
In this notation, we have. 

/ \ 7 *> W A I ^W 9W 

( a ) . dw= -=— ax-f- ^— a y-\- -=- dz 
v dx ' dy l dz 

35. The use of Principle 3 greatly simplifies the dif- 
ferentiation of many complicated expressions. In the 
first place it reduces differentiation to the consideration 



28 A PRIMER OF CALCULUS 

of one variable at a time, and secondly, an expression 
involving one variable only may be made a function of 
several variables by replacing selected component parts 
of the expression by new letters for the time being; 
thence differentiating as to the several variables and 
adding, we find the differential of the whole. Thus 



\-o 



d.x 2 =d.xy, where y=x, =d x .xy-\-d y .xy 

=y dx-\-x dy = xdx-\-xdx = 2 xdx ; 
d.x z =d.x.x.x=x 2 dx-\-x 2 dx-{-x 2 dx=Sx 2 dx; 

and in general, if n be any positive integer, then, 

d.x n =d.x.x... to n factors =x n ~ 1 dx-{-x n ~ 1 dx -f-... 
to n terms = wx n—1 dx. 

It is not necessary to replace each component ex- 
pression that is fixed upon as a single variable by a 
new letter, since a little practice in retaining the idea 
of its singleness of value as distinct from other such 
single values into which the expression may be con- 
ceived as separated, will accomplish the same purpose, 
and be shorter and easier. The above differentiations 
of x 3 and x n are examples of this, in which each factor 
x is conceived as distinct from every other such factor 
for the purposes of partial differentiation as to that 
factor, without the necessity of displacing it by another 
letter for the time being. 

Powers 

36. Rule 1. To differentiate a power with a constant 
exponent, multiply the poiver by its exponent and the differ- 



PRINCIPLES AND RULES 29 

ential of its base, and divide the result by the base. In 
symbols 

d.xP =p.xP dx/x=p xP~ 1 dx. 

The proof of this rule for any real exponent p 
can be made to depend upon the theorem that 
lim (yP — 1)/G/ — 1)=P when lim y=l. This is 

proved when p is a fraction, ±m/n, by dividing out 

1 
the common factor y~n — 1 from numerator and de- 
nominator before putting y=l ; then if p is an incom- 
mensurable number between the fractions q, q', in order 
of magnitude, the quotient (yP — l)/(y — 1) will lie 
between the similar quotients found by replacing p by q 
and q f , and its limit will therefore be between q and q'. 
Thus since this limit always lies between the same two 
fractions as p does, it must be p. Hence 

(yP V) x p 

NA.xP = N(x'P—xP) = ^ -f— NAz, 

where y=x!/x, and therefore approaches 1 as x' ap- 
proaches x. Thus 

d.xP =lim NA.xP —pxP dx / x=pxP~ 1 dx. 

To prove this rule for imaginary exponents, it must 
be necessary to define such powers. It might be made 
a condition of that definition that the above rule should 
be true, and this would, in fact, be sufficient to deter- 
mine such powers in connection with the condition 
that 1p=1. 

37. Generalized Rule 1. To differentiate a product 
of powers, each with a constant exponent, multiply the pro- 



30 A PRIMER OF CALCULUS 

duct by the sum of the products consisting severally of each 
exponent times the differential of its base times the product of 
the remaining bases, and divide the result by the product of 
the bases, i. e., 

d.x m y n zP = x m y n zP (myzdx-\-nxzdy -\-pxydz) / ' xyz 

and for x m y n zP /xyz we may write x m ~ 1 y n ~ 1 zP~ 1 . This 
rule verifies by the application of Principle 3 and Rule 1. 

38. The preceding rule serves to differentiate frac- 
tions, remembering that a denominator appears in the 
numerator with opposite exponent. It is necessary to 
bear this in mind but not to write it so. Thus, 

d~ — — (x dy — y dx) , since — = x — 1 



qjm ym — 1 \ 

^=-^+r( mxd y— n y dx )> since ^ 



39. Rule 2. To differentiate a power to a constant base, 
multiply the power by the differential of its exponent and the 
natural logarithm of its base; i.e., d.a x = a x log adx, 

a^x — 1 

We have A T A . a x =N(a z '—a x )=a x — N A x and 

a h i 

d.a x =\imN&.a x =u x .fa.dx where/a=lim — y — ■ as 

h approaches zero, and we have to show that fa = log a, 
to the natural base e. We will not enter into the proof 
that (a h — 1)/ h has a definite limit depending upon a 
alone as h approaches zero, but take it for granted that 
this is so. How that may be can be seen in a numerical 



PRINCIPLES AND RULES 31 

case by taking a = 10, and h=l, .1, .01, .001, .0001, 
etc., using ordinary tables of logarithms to compute 
the successive values of (10^ — V)/h. The limit of this 
quotient to 6 decimal places should be log 10=2.302585, 
as taken from a table of natural logarithms. 
The definition of e is that 

lim — -j — ==1. 

Thence, if m=loga so that a=e m , we find 
■1 ,. e mh — 1 ,. e mh — 1 

h=o n //=0 mh=0 

Thus fa = log a. 



lim — ; — = lim — ; = m lim - n — = m. 1 = log a. 

h h =Q h m h=o mh 



40. To determine the value of e from its definition, 

let h=-\/n\ then when n is very large, e n = (l-| — ) n 

will be an approximate value of e, since (e n h — V)/h=l, 
while (e h — V)/ h must be very nearly equal to 1 (the 
more nearly so the smaller h is taken i.e., the larger n 
is taken). Thus 

e= ii m (l _|_ I )« = lim [l+7i. ^+ 7l(n ~ X) i-+ 
t^oo n n ' 2! ?i 2 ' 

nCn— 1) (7i—2) 1 

3! n*"*"""- 1 

by considering ti a positive integer and using the 
binomial theorem. But the quotients 

71—1 (71—1) (71—2) 

7i ' n* 



32 A PRIMER OF CALCULUS 

are all smaller than 1 and approach 1 when n approaches 
infinity, so that 

.=1+1+^ + ^ + ... =2.71828... 

When no base is given, as in log a, logx, etc, the 
natural base is to be understood. The logarithm of x 

to the base a is written log a x. We have a ° s °> x =x, 
and thence log a £.loga=logx or log a £=logx / /loga. 

41. As a particular case of Rule 2 we have : 

To differentiate a poiver to the natural base, multiply the 
power by the differential of its exponent, i. e., d.e x = e x dx. 

42. We may combine Rules 1, 2, and Principle 3 
to differentiate a power whose base and exponent both 
vary. Thus, 

d.xy=d x .xy-\-dy.xy=yxydx/x-\-xy logxdy. 
Logarithms 

43. Rule 3. To differentiate the logarithm of a vari- 
able to any constant base, divide the differential of the variable 
by the variable and the natural logarithm of the base, i. e., 
d log a x—dx/xloga 

For put y=loga x ; then x=ay , and dx=ay\oga dy= 
x log ady; or dy=dx/x log a. 

Note the simple form this rule takes for natural 
logarithms, viz., d\ogx = dx/x. 



PRINCIPLES AND RULES 33 

Trigonometric Functions 

44. Rule 4. To differentiate the sine of a variable, 
change to the cosine of the variable and multiply by the dif- 
ferential of the variable; to differentiate the cosine of a 
variable, change to the sine of the variable and multiply by 
the differential of the negative of the variable, i. e., d sin x = 
cos xdx, d. cos x — — sin x dx. 

For A 7 A sin x=N( sin x' — sin x)=2 iVcos ~T' sin — - — 



rf-\-x at * sin (9 . . , ,~ 

:cos — ^ — iVAa; — — where = b.x/l. 



In these functions, the variable usually called an 
angle is, as an actual fact, a number, viz., the number 
of radians in the associated angle, not its number of 
degrees or number of minutes, or number of any other 
units of angle besides the radian, which is 180 /tz 
degrees. Thus 

sin 0/0= chord of arc of 20 radians / arc of 20 radians, 

whose limit, as approaches zero is, 1. Thus, 
d. sin x= cos xdx; and if in this we replace x by 

~ — x, the measure of the complementary angle, we 

find dcosx=smxd (-— x) = — sin xdx. 

45. The remaining trigonometric functions tan x = 
sin x/ cos x, sec #=1/008 2;, vers#=l — cos x, etc., 
may be differentiated by preceding rules and are left as 
exercises, viz. : 



34 A PRIMER OF CALCULUS 

d tan x = sec x 2 dx, dcotx = — esc a; 2 dx, 

d sec x == sec x tan x dx, d esc x = — esc x cot x dx 

d vers x = sin x dx. 

46. Kule 5. 

ds\n~ 1 x=dx/ J(l — x 2 ) = — dcos~ 1 x 
dtan - 1 x — dx/( l-^-x 2 ) = — d cot - x x 
dsec~ 1 x = dx/x s /(x 2 — 1) = — dcsc~ 1 x 
dyers~ 1 x = dx/ J(2x — x 2 ) 

For let y = sin~ 1 x, then x = siny, dx=cosydy = 
J(l — x 2 )dy, or dy=dx/ ,J(1 — % 2 ). The radical is 
positive, because y=^ sin -1 x is a number between 

— 7r/2 and n/2 by convention, in order to make it a 
one valued function of x. This corresponds to an acute 
angle, positive or negative, and thus cos y is positive. 
Also 2/= cos - 1 x is by similar convention a number 
between o and -, so that sin y is positive, and dx = 

— sin y dy = — J ( 1 — x 2 ) dy. The remaining rules are 
similarly proven, and are left as exercises. 

Anti-Differentiation 

47. The phrase "anti-differential of" or, "a func- 
tion whose differential is," is symbolized by the charac- 
teristic d~ 1 . There is another equivalent symbol called 

"integral of" (viz., the old form, I , of the letter s) 

which will be explained under integration. For the 
present, we consider it as another characteristic equiva- 
lent to d~ 1 , and integration as another term equivalent 



PRINCIPLES AND RULES 35 

to anti-differentiation. E. g. , d~ 1 2 x dx=\2 x dx = a 

function whose differential is 2xdx. We know from 
differential results that x 2 or x 2 -\-l or x 2 — 3, or in 
general, x 2 -\-c, where c is arjy constant, is a value of 

J 2xdx; while by Inverse Principle 1, any value of 

this anti-differential (or integral) must be obtained by 
adding a suitable constant to x 2 . Thus while problems 
in anti-differentiation have any number of different 
answers, yet each must differ from any other by a con- 
stant only. 

48. An anti-differential is unambiguously determined 
when we know what is called its initial value, i. e, its 
value for given "initial" values of its variables. Thus 

if we require the value of | 2 x dx that is zero when 

x=o, then x 2 is the only answer, if we require that 
value which is 5 when x=o, then x 2 -\-o is the only 
answer. When an integral is sought corresponding to 
a given differential, the given differential therefore 
determines only what may be called the variable part of 
the integral, and there is some constant to be added to 
this variable part, which the initial value of the integral 
determines. 

49. Anti-differentiation, like all inverse processes, 
is largely a matter of tabulation of the results of the 
direct process. We consider that there is a table of all 
differentiable functions arranged in one column and 
opposite each function in an adjacent column is placed 
the differential of that function. To find an integral of 



36 A PRIMER OF CALCULUS 

a given differential, we look in the differential column 
until we find in it the given differential, when opposite 
such differential in the function column we find a 
required integral; and if, further, we seek a definite 
integral with given initial value, then we add to the 
value found in the table a suitable constant to make it 
satisfy the given initial condition. It is impossible to 
make a complete table of this character. Such tables 
as are made for practical use consist of differential forms, 
each of which covers a great variety of cases. We 
proceed to consider the more important forms, which 
take the place in differentiation and integration that 
the multiplication table takes in multiplication and 
division. 

50. The Power Differential, d.cx m =mcx m - 1 dx. 
In words, the power differential consists of a power with 
variable base and constant exponent not — 1, multiplied, to a 
constant, by the differential of the base. This differential is 
integrated by reversing Rule 1, viz., multiply the power dif- 
ferential by its base and divide the result by the product of 
the increased exponent and the differential of the base. 

E. g., 15 (x 2 — 4x-|-2) 2 (x— 2)dx is a power differential 

3 
whose base is (x 2 — 4x-f-2), and the exponent is -; the 

A 

remaining factor is, to a constant, the differential, 
2(x — 2)dx, oi the base. We therefore multiply by the 

base, which increases its exponent to - and divide by 

5 A 

the product, - . 2 (x— 2) dx, giving 3 (a; 2 — 4x+2) 2 . 



PRINCIPLES AND RULES. 37 

51. The generalized power differential, which comes 
from differentiating a product of powers, is not easy to 
recognize; we can only suspect that a differential is of 
this form, and try multiplying by the product of the 
supposed bases and dividing by the sum of the products, 
each consisting of an increased exponent into the dif- 
ferential of its corresponding base into the product of 
the remaining bases, when the result, if it is correct, 
must be, to a constant, the product of the powers of the 
bases to the increased exponents. 

52. The Logarithmic Differential, d.c log c'x = 
cdx/x. This is the form excluded from power differ- 
entials because the exponent of the base is — 1. To 
anti-differentiate this differential, we multiply it by the 
base and the natural logarithm of any constant times 
the base, and divide the result by the differential of the 
base. The introduced constant c' must be such that ex 
is a real positive number; usually it is 1 or — 1 ac- 
cording as x is positive or negative; and it may be 
omitted altogether, since we know that its introduction 
is equivalent to the addition of the constant c log c f . 

K S- > f-^Zj- =i lQ g O 2 - 1 ) or | log (1— x*}, 
according as x 2 > 1 or x 2 < 1. 

53. The following differentials are logarithmic, but 
their bases are not readily discovered, and it is best to 
include them among standard forms, 

dx dx dx 1 f dx dx ~| 

J(x*±a*)> xj(a*±x*y x*—a*~ 2a~lx^a~x~+^\ 
whose integrals may be verified as, 



38 A PRIMER OF CALCULUS 



loglx+vc* 2 ^ 2 )!, -log 



1 x — a 

54. On the contrary the differentials (similar to the 
above), 

dx dx dx 

V(a 2 — a.' 2 )' xj{x*—a*) m x 2 -fa 2 ' 

are anti- trigonometric forms, their integrals being, 

x. 1 x 1 # 

sin -1 —, -sec -1 —, -tan -1 -, 
a a a a a 

The anti-vers form, ..^ — - = d. vers - 1 — is also 

J {lax — x 2 ) a 

nr n 

the anti-sin form d. sin -1 . In fact the two in- 

a 

tegrals differ by the constant n/2. 

55. Other logarithmic differentials are, 

secxdx, cscxdx, t&nxdx, cot x dx = cos x dx/ sin x. 
By multiplying and dividing the first three by 

sec x -J- tan #, esc a: — cot x, sec x, 
respectively, the integrals are seen to be, 
log (sec x -f- tan x) , log ( esc x — cot x) , log sec x, log sin x. 

dx 

56. The differential — : 7-7 ; — , reduces to 

asmx-\-ocosx-\-c 7 



PRINCIPLES AND RULES 39 

c'dz 
the form _,_ . , according to the values of a. b, 
z l ± c l 

c, by the transformation 2=tan^.t, and therefore 

d 2= J sec \x' 1 dx. To make the transformation, first 

put sin x= 2 sin Jxcos \x, cos x = cos -J- £ 2 — sin^z 2 , 

6=6 (cos J a: 2 + sin J a 2 ) and multiply both numerator 

and denominator by sec \ x 2 . 

57. The integral I „ 2 _i_ .-na ' where w is a positive 

Jc f dz 
„ 2 , , by 
2 — J— C 

successive applications of the formula, 

^ J (z* +C 2 ) ?i = 2(n— l)c 2 | (ga+c 8 )"-' 1 "^ 

( 2 »- 3 )/^y»- 1 } 

This formula may be verified by differentiation. 

fm % 
fn * 

where f m x, f n x are entire functions of x, of degrees ra, n, 
respectively, with real coefficients. If m is not less 
than n, we divide the numerator by the denominator 
to a remainder of less degree than n. The entire part 
of the quotient is integrated term by term under Inverse 
Rule 1. We have therefore only to consider a proper 
fraction of the above form, i.e., one in which m<^n. 
By the theory of equations, the denominator f n x factors 
into real irreducible or prime factors, of the forms 
x — a, or (x — 6) 2 -(-c 2 , each occurring to certain 



58. It can now be shown how to find | : 



dx, 



40 A PRIMER OF CALCULUS 

powers. By the theory of resolution into partial frac- 
tions, the given proper fraction f m x/f n x will reduce to 
a sum of fractions each involving a power of one prime 
factor only in its denominator, and a numerator one 
degree less than the prime factor of the denominator, pro- 
vided, all fractions of this form be included in the sum whose 
denominators are divisors of f n x. The integration of any 
such partial fractions comes under preceding methods. 
The factoring of the denominator, and the resolution of" 
the fraction into the sum of its partial fractions, are 
algebraic problems. 

59. The Exponential Differential, d.ca x = 
c log a. a x dx. This differential consists of a power with 
constant base, multiplied, to a constant, by the differ- 
ential of the exponent. It is anti-differentiated by 
dividing it by the product of the differential of the exponent 
and the natural logarithm of the base. 

60. The Trigonometric Differentials, csinxdx, 
c cos x dx, c sec x 2 dx, c esc x 2 dx, c sec x tan x dx, 
czsaxcotxdx. These consist of certain trigonometric 
functions of a variable, multiplied, to a constant, by the 
differential of the variable. The integration consists in 
replacing these functions in each differential by the 
corresponding function of the same variable from which 
it is derived and dividing the result by the differential 
of the variable, and also by — 1 if the resulting function 
is a "co" function. Thus, the integrals are respectively, 
— ccos#, c sin $, ctanz, — ccotx, csecx, — ccscx. 

61. Integration By Parts. Principle 3 can be 
reversed in integration. Its particular application is to 



PRIN CIPLES AND R ULES. 4 1 

a differential udv, where u is a chosen variable factor 
and dv a known differential. From d(uv) = udv-\-vdu, 

we find j udv=uv — I vdu. In words, to integrate by 

parts, integrate as if a chosen variable factor were a constant 
for the first term, and complete the integration by subtracting 
the complete integral of the differential of the first term as if 
the assumed constant alone varied. The success of this 
method, when it is applicable, depends upon the choice 
of the variable factor, which must be such that the 
new integral to be found is easier of solution than the 
given one. 

E.g. Cx 2 e x dx=x 2 e x — 2 Cxe x dx 

=x 2 e x — 2xe* + 2 Ce x dx 

=x 2 e x — 2xe x + 2e x . 

This is by taking e x dx=d.e x as the known differen- 
tial each time, and the remaining factor as a constant 
in the partial integration. But if Ave take e x as the 
assumed constant, we find 

I x 2 e x dx= — e x — J- j x s e x dx, 

and the new integral is more difficult than the old. 

62. In the following examples that give a differen- 
tial, followed by one or more of its integrals, it is 
required: first to verify the differentiation by the prin- 
ciples and rules of preceding articles, as an exercise in 



42 A PRIMER OF CALCULUS 

differentiation; secondly, to obtain the integral from the 
differential by the inverse rules or methods of the 
preceding forms, as an exercise in such methods and 
rules. The more important integrations may be taken 
as fundamental forms in any subsequent examples of 
integration. 

Examples. 

DIFFERENTIALS. INTEGRALS. 

1. (8x 3 — 9x 2 + 6x— 7)dx; 2x 4 — 3a; 3 +3x 2 — 7x. + 8 

2. (SJ X -\^y — 9x^4-~)dx; 2x^+12 Jx — 3x 3 — x -2 

3. (4x—l)dx; 2x*—x + 3, (4x — 1)»/8. 

4. (4— Zxydx; 3+16£— 12x 2 + 3x 3 , — (4— 3x) 3 /9 

5. JQLa+Vx) dx; 2(4a + 9x)^/27 

6. S(fs/V(a2 + S 2) ; > /( a 2 + 6 .2). 

7. x n - 1 {a^-bx n y i dx; (a-\-bx n ') h + 1 / /bn(h-l r l) 

8. x m (a + 6a; ? ;i \(m + l)a+(m-\-nh+n+l')bx n ]dx; 

x m + 1 Qa+bx n ) h + 1 

9. x- ,? ^+ 1 )- 1 (a + 6a; ft ) /i da;; 

— x-^( h +V (a+bx n ) h+1 /an(h + l) 

10. z 2 ^;/(a 2 + cx 2 )*; £ 3 /(a 2 +^ 2 )^3a 2 

ft +3 k±l 

11. x h dx/(a 2 +cx*)-2-; x^ 1 / (a 2 +cx 2 ) 2 a 2 (/i+l) 

ft /H-2 

12. x(a 2 +6'a; 2 ) 2 cfo; ( a 2 -L-cx 2 ) 2 / c (h + 2) 

J (a 2 4-rx 2 ) 

13. V( a2 +«c 2 )cfa = — — ^^ — • x d%, for integration 

a 2 C dx 

by parts; i^( a2+CT 2 ) + _j _____ 



PRINCIPLES AND RULES 43 

14. J(a*—x*)dx; ixj^—x^ + ^sin- 1 * 

A CL 



15. N / (a 2 +x2) ^ ; l x>/(a 2J_ >T 2 ) + __l 0g( , :+N/a 2 +x 2 ) 

£ ( a 2 #2)2 

16. (a 2 — x 2 ) 2 dx = . x 3 dx, for integration 

v x z 

by parts ; 

17. (a« + 2:2)l^ ; 

f(5a2+2a:2) N / ( a2 + ,;2 ) + ^i og(x+N /^q^r ) 

18 - /^/^ 2 ^ ' — V( '2ax-x 2 ) + a vers" 1 5 

sj{2ax — £ 2 ) a 

iq cfo . V (2a*— x 2 ) 

' xJ(2ax—x 2 Y ax 

20. J(2ax — x 2 )dx; ^^ J^ax—x^+^j-yevs- 1 ^ 
21. 



^V^ 2 - 1 ' 2 )' 



J(a 2 —x 2 ) 1 



h-rrrlog 



2 a 2 a- 2 ' 2a 3 ° a + </(a 2 — x 2 ) 

22 dx J(* 2 -" 2 ) | 1 , cc _^ 

* • x *,J(x 2 — a 2 ) } 2a 2 x 2 ^ 2a 3 a 



ztfo 1 

a 2 -f-c£ 2J c 

'fa 1 , 1 , x r 



23 -^^; jlogV(a 2 + cx 2 ) 



24 - -7 — , 7 n N ; log (a x- n +6), — log — — =— 



44 A PRIMER OF CALCULUS 

25. 1 ±^- dx: lo| 






27 - J^i *"' *+* lo § ^ - ^ 3 tan_1 73 

28 - *«+i ; 472 log ^-,72+1 + 272 tan l±ii 

30. dx/xlog x; log 2 cc = log log a; 

31. log Ecfa/av log a; 2 / 2= log £. logx/2 

32. 2xdx/x*\ log. a; 2 

33. .— — ; I . (a function not tabulated) 

log x J log £ 

34. J^pdx; 

v b-\-x 

V(a+aO(& + z) + (a— 6) log ( Vtt+z + Jb+x) 
[Fut b+x=y 2 , dx—2ydy] 

35 ' V^- dx;J(a—x) (6 + x) + (a + 6) sin-y5±£ 

36. (e^+e-^) 2 ^; \ (e 2x — e~ 2x + 4x i ) 

37. f!zil dx; 2 log (e* + 1) — x 
e x -4-1 

38. a x b x dx; . a x b x / log a b 



PRINCIPLES AND RULES 45 

3g> £_^_ l e 2xJ re .vlJ r l og ( e z_ 1 - ) 

40. x 2 e x dx; e x (x 2 — 2x + 2) (int. by parts) 

41. a: 2 logxdxy -Jx 3 (31ogx — 1) 

42. sin - 1 xdx; scsin —1 x -4-^/(1 — x 2 ) 

43. sin 2 x. dx; — ^- cos 2 x 

44. cos x 2 dx; i x -\-i SU1 2a; 

45 . cos x 3 dx ; sin x — J- sin x 3 

46. (tan x-(-cotx) 2 dx; tanx — cotx 

47. (tan 2x — 1) 2 dx; \ tan 2x + log cos 2x 

48. e mx siii'ttxdx; e mx (msinnx — ?icos?ix) / /(m 2 -4-n 2 ) 

[Integrate twice by parts with e m x as constant] 

49. e m x cos n x dx; e m x (n sin n x-\-m cos nx)/ (m 2 +w 2 ) 

50. tanx 2 dx; tanx — x 

51. tan x 3 dx = sec x 2 tan x dx — tan x dx ; 

J tan x 2 -(-log cosx 

/tan x n— * /* 
tan x"dx= 1 ( tan x n_2 dx 
n — 1 J 

53, tanx 2?l dx, n a positive integer; 

tanx 2 "- 1 tanx 2 "- 3 n , +1 , . 

is=i ^=-^+-+ ( - 1)+(tanM) 

54. tanx 2 " - * dx, n a positive integer; 



tanx 2 " -2 tanx 2ft — 4 , . . . . tanx 2 , . 
~2S=2 ^ =I -+-+(-l)"(- T - + logcos.) 

55. secx 4 dx=secx 2 (l-(-tanx 2 )dx; tanx-|-Jtanx 3 



46 A PRIMER OF CALCULUS 

56. cosx 4 dx=^ (l-f-cos2z) 2 dx; 

g 1 ^ (12 x -\- 8 sin 2x -f- sin 4x) 

57. cosx 4 sinx 2 <ix=-|(l + cos 2x) (sin2x) 2 cfa 

= yV (1 — cos 4.x) dx-{-^ sin 2x 2 cos 2x dx; 

x sin 4x . 1 . _ o 

i6 — 6i~+^ sm23:3 

58. cosx 4 sin a 3 c?x = cos x± (1 — cos a: 2 ) s'mxdx; 

— -J- cos x 5 -f- 1 cos x 7 

59 - 5-4ts2, ; Jtan-i(3tan.) 

60. If sinhx =%(e x —er x ) 
cosh aj= J (e* + e — * T ) 
then coshx 2 — sinha; 2 =l 
d sinh x— cosh x dx 
d cosh x=sinh x dx 
dsmh- 1 x = dx/ J (x 2 + a 2 ) 
dcosh - 1 x = dx/ J (x 2 — a 2 ) 

63. Reduction Formulas 

Let v—J(a 2 -\-cx 2 y, then we have the differential 
rules, 

d.v n = cn v n ~ 2 xdx 

d \ - \ =a 2 n - — -. 

lv J lv J v 3 

These rules may be used to integrate x m v n dx by parts 
in six different ways, so that the new integral shall be 



/ 



x mf v n ' dx where m\ n' are one or both two units 



smaller than m, n; and repeated applications of such 
integrations will thereiore reduce the given integral 



RED UCTION FORMULAS. 47 

eventually to dependence upon standard forms, either 
algebraic, logarithmic or anti-trigonometric, when m, 
n are any integers positive or negative. These formulas 
are, 

J x m v n dx = 

(a) \v n .x m dx 

_ | £ m + l V n _ c ^ f ^ m + 2 ^n-2 ^ 1 

(b) Cx m ~\ xv n dx 

= — - - \ x m - 1 v n + 2 — Cm—l) Cx m - 2 v n + 2 dx X 

= — ■ \x m + 1 v n 4-a 2 n Cx m v n ~ 2 dxX 

m-f-n+1 (. J j 

(d) (V+»+s. J?!_ <&. 

= — — { — x mJ ^ v n + 2 -4- Cm 4- n 4- 3) f x m v n + 2 dx \ 

/^ m — 1 
v m-l 

v n+2 a 2( m _l) \x m - 2 V n dx X 

c(m-\-n-\-l) ( J ) 



m+n+3 



x m — L V 

x m dx 

^m + 'S 
x m+l v n+ 2 — C( 



(f) JV 

j ( m -j. n +3 ) (V+2 i>* da; } 
a 2 (m-j-1) ( J ) 



48 A PRIMER OF CALCULUS 

In these formulas, a 2 may be changed to . — a 2 
throughout, and c is usually 1 or — 1. 

Similarly, let u=sinx, v— cosa;, so that 

d.u n = nu 71 —^ dx, 
d.v n = — n v n ~ 1 u dx 

7 (u~\ n u 71 - 1 dx 7 fvV l v 71 - 1 dx 
d. - =n. :T — , a - == — n — 

and then 

(u m v n dx=: 

(a') lv n - 1 .u m vdx 

== — — ■ i u m+1 ^- x + (w — 1) f u m+2 v n ~ 2 dx 
ra + 1 ( J 

1 

— n+1 

(C) J^- 

= — : — j w m+1 v n - 1 +(n— 1) (V*^- 2 cfa 1 
m-\-n ( J ) 

( d') jf — - ^ 



. w r»-i v»+i_j-(m — 1) \u m ~ 2 v n+2 dx 



,71 — 1 

u m+n-l v fa 



m+n + 2 



W+l I 



- n m+1 v n+1 + (m+n+2 ) Cu m v n+2 dx 1 






m+w -%dx 



REDUCTION FORMULAS. 49 

1 



m -\- n 
(f) Cv m+n+2 . 



v n+l _L. ( m _ 1 ) C u m-2 v n fa 



,m+2 



= — — 1 w ™+V>+i-L.(m + n-f2) fw m+2 ^ dx | 

61 . sin x 6 dr = tan x 5 . cos x 5 sin x dx; 

— -J- tan x 5 cos x 6 -f- f I tan a; 3 cos x s sin .r dr, 

— J tan x 5 cos x 6 — -£% tan x z cos z 4 + f I tan x cos x sin x dx, 

hx 
— ^ sin x 5 cos a: — T 5 ¥ sin x 3 cos x — T 5 g sin x cos x -f- — . 

62. cos x 4 sin x 2 dx; 

cos x sin x ' x 

To ■ (3 + 2 cos x 2 — 8 cos a- 4 ) + — - 

63. -, ; (x—. + ^— : 4 sm x) 

smx 4 cosx 3 cosx 2 dsmx 3 ' dsmx * 

-(— | log (sec cc -(- tan a;) 

64. Indeterminate Forms 

fx 

Rule. To evaluate ■=- /or a ^u*en value of x that makes 

fx=o, Fx—Oj differentiate both numerator and denomina- 
tor, before substituting the given value of x; and similarly 
for values of x that make fx=co, Fx = oo . Before making 
such differentiations, any factor of f(x)/Fx that is not 
zero or infinity for the given value of x may be replaced by 
its value for such value of x. 



50 A PRIMER OF CALCULUS 

For let fa = o, Fa = o, then: 

fa .. fx' .. fx' — fa _. A fa dfa 
—- =hm jjj-, = hm ■£-/ — -yr- = lira — n- = vW- . 

ta x > =a Fx' x > =a Fx'—Fa A Fa dFa 

If/a = cc, Fa = co , then l//a = o, 1/Fa^o, and 

o = it- = l im rr/7-= hm -p^— / -f— - 
Fa jPo; /a Fx 2 /x 2 

= b 2 dFa/dfa; i.e., b = dfa/d Fa. 

Finally, any factor whose limit is finite, can, by the 
principle that the limit of a product equals the product 
of the limits of its factors, be at once replaced by its 
limit, and the limit of the remaining factor may be 
found by itself. 

Exponential indeterminate forms must be evaluated 

through their logarithms. E.g. ?/ = (l-| — ) x when 

x — go whose form is l 00 , must be evaluated from 

logy — j log (1 + - )=log (1+z) / z where z =-=o„ 

This is o/o, and is therefore when z=o, d log (1+z) /dz 

= _-='l ; log2/=l, y=*. 

64. Evaluate the following functions for the given 
values of x : 

( a ) L ^zr> x=1 (V 2 > 

T 3 r 2 r _L_ 1 

0» sqdbsri' * =1 ® 



INDETERMINATE FORMS 51 

(d) log^/Cx— 1), z=l (1) 

(e) (e^+e-^/x, z=o (2) 

a: sin x— a: 2 

< f) 2~ oss + s'-^ ' 3 ^ C ~ 2) 

(g) log sin x /cosx, £=tt/2 (0) 

(h) sin - 1 x/s\\\x, z=o (1) 

(i) tan a:/ log cos x, a*=-/2 (oo ) 

( j) ^(cosx— l)/zlog(l+.r), x=o (— J) 

[Xote that the factor e :( * can be replaced by e°=l 

before differentiating numerator and denominator] . 

(k) secx — tan x, x — -/2 (0) 

(1) (l+x 2 )% x=o (1) 

± 1^ 

(m) (1+x) * 2 , £=o 0,0) 

i 

(n) (1-f ma;) 1 , x=o (e m ) 

(o) (log*)*, x=o (1) 



(p) £ log(1 -^, a-=x (e) 

65. Applications of Inverse Principle 1 

65. If xdx-\-ydy=o and 2/ =a when x=o } 
then x 2 +?/ 2 = a2 



A PRIMER OF CALCULUS 

66. If ~j--\-^—^-=o and y=b when x=o, 

x 2 v 2 
then ^ + ^ = 1 

67. If -r + -r= o, and ty=a when oj=o, 

then a^ -f- 2/3 = a% 

68. If -t^= — , and ?/=o when a; = o, then2/ 2 =4aa; 

rd# - 

69. If — : — —c, and r=a when ^=0, then r=a e c 

clr 

70. If (l+2-) 3 =l+3x+3x 2 +a; 3 , then by integration, 

(l-f:r)4=l+4£ + 6x2 + 4x3-fx4 

71. From,-i T ==l-f-a; 2 +a; 4 +a;6+... ', (x 2 <l) 

show that ^-^ = 2 (z+i^+ix 5 + ... ), 

72. From 



-l +lr8 4. U r 4. I 1.3.5... T 2.-1) 

when a* 2 <l, show that for the same values of a;, 
sin^ 1 a; = 

-LiEi'J-t? — 4 1.3.5... (2n—l) ? 2 " +1 

X+2 3 + 2i5 + "' ,+ 2.4.6.. .2ti 2?i+l + "* 

73. From-i-^ = l— x 2 +x±— &*+.„ (x 2 <l) 

show that, tan - 1 x—x — ^x 3 -\-^x 5 — ... 

and that, tan -1 .-2-^ = - 4-£ — i-£ 3 4-i-a; 5 — ... 
' 1 — x 4 ' rf ' & 



APPLICATIONS OF INVERSE PRINCIPLE 1 53 

* 

74. Show that the function e x defined by the differen- 
tial equation d . e x =e x . dx, and the initial value e°= 1, 
satisfies the exponent law e x . ev=e x+ y, and that this 

x 2 X s 
function is e x = 1 -f- x + — + + + • • • ( a convergent 

series for all values of .r). 

[Prove the ratio of the two members constant, by 
differentiating such ratio, and determine the constant 
by its value for x=o] . 

Note. e^= 1+1+ — + -^- + ..= 2. 71828... =e; also 

by the exponential law, e 2 =e 1 .e 1 —e.e J e*=e 2 .e 1 =e.e.e. / 
e i —e 3 e l =e.e.e.e ) etc. Thus the exponent notation e x 

x 2 
for 1+.T+ — + .. agrees with the usual meaning ol 

this notation when a* is a positive integer, and it must 
agree also when x is any real number, since the gener- 
alization of the exponent in elementary algebra, is 
derived from the exponential law, a m . a n =a m+n . The 
above generalization extends also to imaginary ex- 
ponents. 

75. If 1=^ — 1, show that e ix =cosx-\-isinx 
tf-^=cos x — i sin x, and thence 

%2 r 4 r Q 

cos x = i (e™ + e~ lx ) =1 — — + — — -gf +» ■ 
1 a; 3 cc 5 

sina;= 1H ( ^ x — ^~ l:r ) = ^ — 3T + 5T — • • * 

[Prove e~ ix (cos a; +^ sin a;) = constant = 1] . 



54 A PRIMER OF CALCULUS 

Note. The above results give general definitions of 
cos x and sin x for all values of x, imaginary as well as 
real. The exponential values of cosx, sinx show that 
in all cases cos x 2 -|-sin x 2 = 1, 

sin (x -\- y) = sin x cos y -\- cos x sin y y 
cos (x-\-y) — cosx cosy — sinx sin?/, etc. 

Also, dcosx= — sinx dx, d.sinx=cosx dx coso=l, 
sin 0=0. 

76. If x, y be functions of 0, such that dy=xd0j 
dx= — ydd, and x=l, y=o when #=o, show that 
x=cosO, y=smO. 

[ Prove x cos -{-y sind = constant = 1, and 
x sin — y cos = constant = o, and solve for x, y] . 

66. Expansions in Series 

77. If tan x can be expanded in ascending powers of 

x s 2x 5 
x, show that tanx = x -f- — -) — r-^- + . . . 

o lo 

[Put tanx = ^4 + Bx + Cx 2 + Dx 3 + Ex* + i^x 5 +... 
and determine the co-efficients by successive derivation 
and x = o] . 

/8. If the following functions can be expanded in 
ascending powers of the variable given, then the ex- 
pansions are: 

(a) log(l+x)=x— i-x 2 + J-x 3 — |x 4 -f... 

(b) (1 + X)P 

= l +P x + ^x 2 + ^]^ 



EXPANSIONS IN SERIES 55 

2x3 



(c) e x secx = l-\-x-\-x 2 — 



(d) log cos x— (— +— + — +..)=logsin(-— x) 

Note that log a;, log sin x, cot a:, log cot x, etc., are 
functions that cannot be expanded in ascending powers 
of x such as A -f- Bx -\- Cx 2 + . .. 

(e) Maclaurin's Theorem. fx= 

rt'2 /)" 3 rpll 1 

fo+fo.x+ro.%i+f'"o.^+...+fWo. 



2! ' J 3! ' lJ ' (n— 1)1 ' 

where fx, f'x, f"x, ... p l ~ x h, .. are the successive 

derivatives of fx. 

Note. This expansion certainly fails when fx, fx, 

fx, f"x, ..., are not all finite continuous functions 

for every value of x between o and its final value used 

in the expansion. When these derivations are all 

finite and continuous from o to x, it can be shown 

(Article 68) that the difference between fx and the first 

x n 
n terms of the series is exactly f^x'. — , where x' is 

some number between o and x. Although x' cannot be 
determined otherwise than that it is between o and x, 
yet this form of the difference enables us to assign a 
superior value of it by using the largest value of f^x' 
as x' changes from o to x, and if this superior value 
approaches zero as n approaches infinity, then certainly 
(e) must be true. 

(f). Taylor's Theorem. 



56 A PRIMER OF CALCULUS 



[The remainder after n terms isf n h'. ~ . (x f between 

a;. and x-\-y)~\. 

(x — a) 2 



(g) fx=fa+f'a(x — d)+f'a. 



n\ 
a and x ] . 



[Remainder after n terms =f n \x'. - — , x' between 



Note. From (g) show that if a is a root of fx—o 
then x — a is a factor of fx and conversely. Denning: 
a is an n-multiple root offx=o when (x — a) n is a factor 
of fx, show that the double roots are common roots of 
fx=o, f'x=o, and that x — a (if a is a double root) is 
a common factor offx, fx. So, triple roots of fx — o are 
roots of the greatest common factor of fx, fx, fx, and 
so on. 

67. Maximum and Minimum Values 

fx being a finite continuous function of a real variable 
£, it is said to be ivcreasing for a given value of x when 
its value increases as x increases from its given value and 
decreases if x decreases; and it is said to be decreasing 
when its value decreases as x increases from its given 
value, and increases if x decreases. Also fx is a maximum 
value when for immediately less values of x, fx is in- 
creasing, and for immediately greater values of x, fx is 
decreasing; while fx is a minimum value if it is similarly 
changing from decreasing to increasing at the given 
value of x. The greatest value of fx is necessarily a 
maximum value, and its least value a minimum value. 



MAXIMUM AND MINIMUM VALUES 57 

but there may be other maximum and minimum 
values. 

In other words, maximum and minimum values of 
fx are only greatest and least values oifx for values of x 
in the immediate neighborhood of the given value, both 

dfx A fx d fx 

greater and less. Since — = — =lim — f— , therefore —f- 

dx Ax dx 

A Fx 

must have the same sign as — — for small values of Ax, 

° Ax 

supposing that — ~ is neither zero nor discontinuous. 

Hence : 

(a) fx is increasing when fx is positive. 

(b) fx is decreasing when fx is negative. 

(c) fx is a maximum when fx is changing from posi- 
tive to negative, i.e., when fx is positive for values of x 
immediately less than its given value, and negative for 
values of x immediately greater than its given value. 

(d) fx is a minimum when fx is changing from nega- 
tive to positive. 

(e) A maximum or minimum value of fx can only 
occur for a value of x that makes fx zero or discon- 
tinuous. 

For, if fx is not zero or discontinuous, it will be posi- 
tive or negative, which is case (a) or (b). The most 
common discontinuity is/'x=oo . 

Find the maximum and minimum values of the fol- 
lowing functions: 

78. 2/=5-f-8x— x 2 . 



58 A PRIMER OF CALCULUS 

dy 
[-y- = 2 (4 — x), which changes from -f- to — at a; =4, 

i.e., it is positive when x<4 and negative when x^>4. 
Therefore y=5-\-32 — 16 = 21 is the maximum value 
of y. Also y can be as much less than zero as we please 
by taking x large enough.] 

79. 2/=4 + (x— 3)§ — (x— 3)1. 

I~— ^- = — f^t; which is zero for £=17/5, and 

L ax 6{x — 3)* 

discontinuous for £=3. At £=17/5, -^-changes from 

-J- to — and y is a minimum; at cc=3, -7- changes from 

— to + and y is a maximum. Also 7/ can be as great 
as we please by taking x enough less than zero, and y 
can be as much less than zero as we please by taking x 
great enough, so that both the maximum and the 
minimum values of y, are only with reference to ad 
jacent values.] 

80. y=a sin x-\-b cosx. 

\_-j- =6sina: — acosx; tsaix=a/b, y= ± J '(a 2 ~{-b 2 ) . 

— -j- = — (a cos x-\-b sin x) = — y. Therefore, when 

(XX (XX 

y is positive, -j- is decreasing, hy (b) with -j- in place 

dy 
of y, and remembering that ~ is zero for the given 

value of x, therefore it changes from + to — or 



MAXIMUM AND MINIMUM VALUES 59 

y—sla 2 -\-b 2 is a maximum value. When y is nega- 
tive ~ is increasing, and being zero, it is therefore 

changing from — to -\-, and y= — J(a 2 -{-b 2 ) is a 
minimum value. These are true greatest and least 
values of y, since y cannot increase or decrease in- 
definitely.] 

81. y=(«+l)'» (x— 3)3. 

[min., a5= — 1; max., x=2; min., a; = 3]. 

82. ^=x 2 +i/ 2 , where Zo:+m2/-|-n=o. 
[it? =w 2 / (7 2 +™ 2 ), a minimum] . 

83. Find the largest rectanglar area that can be en- 
closed by a boundary of 200 feet. [2500 sq. feet.] 

84. Find the largest rectangle that can be cut out of 
a circular sheet 6 feet in diameter. [18 sq. feet.] 

85. Find the altitude of the maximum rectangle that 
can be cut from an isoscles triangle, one side being part 
of the base. [-J altitude triangle.] 

86. Find the altitude of the maximum right cone 
that can be inscribed in a sphere of radius a. 

[Let a + a; = altitude, y= radius of base = J(ci 2 — x2 )> 
x = ^a for required maximum.] 

87. Find the basin of largest volume, round or 
square, that can be made with a given number of square 
feet of tin. [Width = double the height]. 

88. A Norman window consists of a rectangle sur- 
mounted by a semicircle. Given the perimeter of the 



60 A PRIMER OF CALCULUS 

frame, what dimensions give the window that will 
admit most light. [Height = width]. 

89. How far must one stand from the base of a 
column to obtain the largest angle of vision of a 
statue on the top. 

[The distance is a mean proportional between the 
entire height of column and statue, and the height of 
the column] . 

68. Remainder in Maclaurin's Theorem. 

Lemma. If Fz, and its derivative F'z, be real and con- 
tinuous functions of the real variable z from z=a to z = b, 
and if Fa—Fb, then will F'z be zero for same value of z 
between a and b. 

For, when z changes continuously from a to b, Fz 
must in the beginning either increase from the value 
Fa, or decrease from that value, and since it returns to 
the same value (Fb = Fa) in the end (and does so by 
continuous change of value), therefore there must be 
an intermediate value of z at which Fz changes from 
increasing to decreasing or from decreasing to increas- 
ing. Let z==x' be such intermediate value of z ; then 
by Art. 67, Fx' is a maximum or a minimum value of 
Fz, and therefore F'x' is either zero or discontinuous, 
and since the possibility of discontinuity is excluded 
by supposition, therefore F'x' =o. 

Theorem. The remainder in Maclaurvrfs theorem after 

% n 
n terms is f^x'. —r, where %' is some value between o and x. 



REMAINDER IN MACLAURIN'S THEOREM 61 

x n 
For, let such remainder be E-. so that E is that 

nl 

function of x which is given by the equation 

(a) fx= 

/v 2 <y 71—1 rfU 

fo+fo.z+ro. $ n +... +/<»- 1 >o. -g-^ + R-^ 

The conditions to which fz and its derivitives/'z, f"z, 
...f( n )z, must comply are that they are all real and 
continuous functions of the real variable z, from z=o 
to z—x) and we can then therefore make up from these 
functions, the following function Fz which, with its 
derivative F'z, are also real and continuous from 
z = o to z = x: 

Fz=/z+ (*__,y,+£=*>!ri + ... 



(n — 1)! ti! 

since preceding terms all cancel. 

But Fz=fx when z=o, in consequence of the value 
of E given in (a); and Fz=fx when z=z, in conse- 
quence of the vanishing of every power of x — z. Hence, 
by the lemma, F'z =o for some value of 2, between 



62 A PRIMER OF CALCULUS 

and x, say z—x'. Substituting this value of z in F'z 
and dividing out factors not zero, we find R=f( n )x r . 

x 2 x n 

90. Showthat the error of log (l-\-x)=x —--^-...zp — . 

is between a^+Vfa + l) and x n + 1 y(n + l) (l-\- X ) n + 1 , 

so that when x is positive and not greater than 1, 

x 2 
log (1 + ^) = ^ 5~+--- How many terms of 

log 2 = 1 — \-\-\ — 4+--- must be taken to compute 
log 2 to an error between .0001 and .00005? 

91. Show that the errors of 

/y*3 /y*27l~—±. 

sin *=*-Bl+- *(2^=)!' 

x 2 x* n — 2 

cosx=l — yj- + ••• =F '(2 n _2)! ' 

are ± sin0£.£ 2? V(2™) !, +cosdx.x 2n - 1 /(2n—l)l 
where 6 is some number between o and 1. Show that 

( x 2 "i n (x^ ^ n ~^ 

these errors are smaller than I — / n !, x ! — | /n !, 

I w J [n] 

respectively; and that therefore, however large x may 
be, n can be taken large enough so that the above 
appro ximatons to sinx, cos x are as accurate as we 
please. How many terms of these series must be taken 
to compute cosl, sinl to errors certainly smaller than 
.0000001 ? 

92. If fx=x n -\- a 1 x n — 1 -{-... -\-a n —iX-\-a n , where n is 
a positive integer, and a 1? a 2 , ... a n , are real numbers, 
show that between two real roots of fx—o, lies at least 
one real root offx=o. 



CONCRETE REPRESENTATION 63 



CHAPTER III 

Concrete Representation 

f>9. Algebraic quantities are represented by concrete 
quantities such as length, area, volume, etc. Negative 
numbers are represented only by the assignment of 
opposite characters of measurement, and then a negative 
measurement of one character means the corresponding 
positive measurement of the opposite character. E. g. , 
— 2 units to the right = 2 units to the left, — 3 units 
up = 3 units down, — 4 radians counter clockwise = 4 
radians clockwise, etc. Imaginary numbers can be 
represented by directed lengths in a plane in accordance 
with the principle that J — 1 denotes change of direction 
through a counter-clockwise right angle, as J — 1 units 
to the right =1 unit up. This is applicable when the 
concrete quantities are such as forces acting at a point, 
but not for ordinary lengths, or areas or volumes. 

70. The differential of a variable quantity must be 
a quantity of the same kind. In fact, the change of 
value, the proportional to this change of value, and 
consequently its limit the differential, must be the same 
kind of quantities as the given variable. In other words, 
the differential of a length is a length, of an area, an 
area, of a force, a force, etc. Concrete representation 
of variable numbers will therefore give corresponding- 
representations of their differentials, and the determina- 
tion of the differentials from the variables is important 
not only for its applications to concrete problems, but 



64 A PRIMER OF CALCULUS 

also because it gives concrete ideas of differentiation 
that illustrate this algebraic process and its principles. 

71. Let OX, OY (Fig. 1) be horizontal and vertical 
axes of reference in the plane of the paper. A variable 
point P in this plane is determined by two variables 
x, y called its co-ordinates, which are respectively the 
measures of the distances of P to the right, and up, from 
the axes. Negative measures in these directions mean 
positive measures in the opposite directions. The first 
co-ordinate is called the abscissa of P, and is OL=x 
units to the right (or briefly OL=x); the second co- 
ordinate is called the ordinate of P, and is LP=y units 
up (or briefly LP=y). If y be a definite real function 
of x, this means that each value of x gives one and only 
one value of y, or that P is represented on each vertical 
line by one and only one point; if y=fx be a continu- 
ous function of x, then the locus of the point Pis a 
continuous curve, crossing each vertical line not more 
then once. As an example of discontinuity find the 
locus of P from x = l to x=3 when y = x -\- integer 
part of x. Conversely, any continuous curve drawn 
from left to right, and crossing each vertical line once 
only, would, if we understand that P always lies on 
this curve, make y a definite function of x. In Figure 1 
the curve drawn is actually a circle of center C, and 
vertical radius AC. The upper half of this circle cor- 
responds to a different function of x from the lower 
half. With a certain unit of length, we have OB =8, 
BC==9, AC=5. For P=(x,y), any point on this 
circle, we find from the right triangle on CP as 



CONCRETE REPRESENTATION 65 

hypothenuse, with sides parallel to the axes, that 
( 2/ _9)2_|_( a; _8|2 = 25 3 i.e., y=d±j2o — (o: — S)' 2 
are the two functions in question. 

72. The curve y =fx is smooth when it has a definite 
tangent FT at each point P, and when the direction of 
this tangent changes continuously for continuous varia- 
tion of P. The tangent at P is defined as the limiting 
■position of the indefinitely produced chord PP' as P' ap- 
proaches coincidence with P. This condition of smooth- 
ness is in fact the condition that fx is differentiable and 
that such differential, fx dx, is a continuous function. 
Continuity and smoothness are implied conditions on 
all curves. There may be exceptional or singular points, 
in this respect, but the continuous changes of value of 
the independent variable that are considered in general 
statements must not include such singular points. 

73. In a given curve there are other functions of the 
abscissa x, of P, besides the ordinate y. Thus, let A be 
an assigned initial position of P on the curve, and let 
the tangent line PT and the normal line CP (perpen- 
dicular to the tangent) meet OX in M, N, and also 
meet a perpendicular to OP through in M', N'. 
Then : 

the arc of the curve is the arc AP=s; 

the ordinate area is the area ABLP=u (described by 

the ordinate) ; 
the slope angle, is the angle XMP= <f> radians ; 
the slope is tan <£ ; 

the tangent and normal (lengths) are, MP and PN; 
the subtangent and subnormal are, ML and LN; 



66 A PRIMER OF CALCULUS 

the polar radius and angle are OP—r and <X0P=# 

radians ; 
the polar radius area is the area OAP= v; 
the polar slope angle and slope, are < OPM =i// radians, 

and tan ^ 
the polar tangent and normal, are J1PP and PiV 
thenar subtangent and subnormal, are M'O and OiV'. 

74. The co-ordinates r, are _po?ar co-ordinates of P; 
the unit of measure for the angle is a counter-clockwise 
radian; the unit of measure for r is the unit to the right 
turned through the angle 0, so that it is in the direction 
0; r is therefore positive or negative according as the 
direction is towards or from P. We generally suppose 
6 taken so that r is positive. The unit of measure on 

M'N' is in the direction — — . The units on tangent 
and normal are in the direction <j> and <^> -|- - ; the 

direction <f> can be taken as the direction of increase of s. 
The area described by the ordinate y is divided into 
positive and negative parts determined by the product 
of the sign of the value of y and its positive or negative 
direction of motion along OX. The area described by 
the radius vector r is also divided into positive and 
negative parts according to its positive (counter-clock- 
wise) or negative direction of turning about 0, whether 
r is positive or negative. Conventions of sign are for 
defmiteness of general statements, i. e., with such con- 
ventions, general theorems can be made holding for any 
position of P on its given locus, that must otherwise be 



CONCRETE REPRESENTATION 67 

separated into several distinct theorems depending upon 
the position of P. In other words, results that are 
obtained from a construction in which all the quantities 
are positive will hold for any possible construction 
when proper conventions of sign are used to interpret 
the quantities, which would not thus generally hold when 
magnitude only is concerned. This is because con- 
ventions of sign make continuously varying quantities 
change from positive to negative when their magnitudes 
are to change from additive to subtractive, as the posi- 
tion of P changes continuously. 

75. To construct the differentials of abscissa x, or- 
dinate y, and arc s, of a given curve, for assigned values 
of £, dx. 

Let P be the 'point on the curve whose abscissa is x; take 
PR—dx units to the right; draw the tangent at P, and 
draw RS parallel to OY to meet this tangent in S; then is 
RS=dy units up, and PS=ds units in the direction of 
increase of s. 

For, let P' be the point on the given curve whose 
abscissa is the new value x'; let the new ordinate 
y' = L f P meet the line PR = dx at Q; lay off on PR the 
length PR' = N.PQ; draw R'S' parallel to OY to meet 
the chord PP' in S'. Then by similar triangles, 
R'S f =N.QP\ and since PQ = Ax, QP' = Ay, there- 
fore, PR'=N.Ax, R'S'=N.ky. The differential pro- 
cess 

lim NAx=dx, or lim PR'=PR, 

consists in making Q, and therefore P', approach coin- 
cidence with P, while making N correspondingly 



68 A PRIMER OF CALCULUS 

increase so that the point R' approaches coincidence 
with R. Two constructions are shown in the figure, 
the first is lettered as described and iV— 3, the second 
is unlettered, Q is nearer to P than in the first con- 
struction, R' nearer to R, and N=7. We are to 
imagine a series of such constructions, unlimited in 
number, in which Q is taken nearer and nearer to P, 
with the object of determining the limit of S' knowing 
that the limit of R' is R. It is easily seen that S' 
approaches S, for, in the first place, R'S' is by con- 
struction always parallel to Y, and therefore its limit- 
ing position is a line RS parallel to OY, and secondly, 
since P' approaches P, and S' lies by construction on 
the chord PP' (produced), therefore S' must approach 
coincidence with a point on the tangent at P, which is 
by definition the limiting position of the chord PP' 
produced. Hence dy—\\m iVAgr=lim R'S'=RS, where 
RS is a line parallel to OY, and meeting the tangent PT 
in & 

Next produce the chords from P to each point of the 
arc PP' = As, in the ratio N:l, and let arc PS' be the 
curve in which such extended chords terminate. The 
arc PS' is then similar to arc PP' by construction, and 
its length is N. arc PP' = N. As; both arcs have also the 
same tangent PT, since the tangent is determined by 
the limiting position of the s amc chords produced, in 
either case. Thus when P' is so near to P that the arc 
PP' is always between its chord and tangent and of one 
direction of bending throughout, the similar arc PS' 
must lie between its chord PS' and tangent PS, and be 
of one direction of bending throughout. Hence as P' 



CONCRETE REPRESENTATION 



69 



approaches P, the arc PS' must approach point to 
point coincidence throughout with the straight line 
PS, since PS' does so; i.e., 

ds = lim NAs = lim arc PS' = PS. 

Observe that the two triangular figures PQP', PR'S', 
each with an arc side, are similar figures, with P as 
center of symmetry, and N as ratio of similitude. Since 
PR'S' approaches coincidence with the triangle PRS, it 
FIG. i. 




appears that the difference figure PQP' approaches 
similarity with the differential triangle PRS, as its sides 
indefinitely diminish. The difference figure reduces to 
the point P; we have left, however, in the triangle PRS 
what might be called its ultimate form. 

(a) The limiting ratio of any arc to its chord as the arc 
is taken smaller and smaller, is unity. 



70 A PRIMER OF CALCULUS 

For, PP': arc PP'=PS': arc PS', whose limit is PS:PS= 1 

(b) The slope of the curve y=fx at the point (x, y) is 
dy / dx=f'x. 

For, the slope is tan <I>=RS/ PR =dy/dx. 

(c) Inverse Principle 1. If dw=o identically, then 
iv = constant. 

[Or, if du=dv identically, then d(u — v)=o identi- 
cally, and u — v=constant.~\ 

For the slope of y=fx being dy/dx, then if dy=o 
for a particular value of x, at such point (x, y) S coin- 
cides with R, or the tangent is parallel to OX; if dy = o 
for every value of x, then the tangent is parallel to OX 
at every point, and the curve y=fx must be a straight 
line parallel to OX, so that y=fx remains constant as 
x changes. This proves the principle for real functions 
of one real variable. More generally, if w be a real 
function of real independent variables x, y, then 

dw,= 77 r- dx-\--p^-dy, can be identically zero only when 

— -=o 5 — -=o, identically, since dx, dy, are arbitrary 

values. Thus w=f(x,y) is constant when either x or y 
changes alone, whatever value the other variable may 
have, and it must then be constant when both vary, 
since */(&, y) =A iB /(x,y) + A y /(a/, y)=o+o=o. 

Similarly for real functions of any number of real 
variables, and this includes imaginary variables, 
regarded as depending upon their real components. If 
w is imaginary, it is w=w 1 -\-w 2l J — 1, where w Y , iv 2 



CONCRETE REPRESENTA TION 7 1 

are real, and dw = dw 1 + div 2 . J — l=o only when 
dw 1 =o, dw 2 =o, so that w lf w 2 , and therefore w, will 
be constant, if dw=o, identically. 

76. To construct the differentials of the polar radius 
and angle, r, 0: 

Draw from P=(r, 0), the length PS= ds on the tangent 
at P, (Art. 75); draw SR X perpendicular to the polar radius 
OP at R,; ivith center draiv the arc PS X equal to R,S in 
length; then is 

PR,=dr, H 1 S=rdd, <POS 1 =d0. 

For take on the given curve, the point P'=(r',0'), 
and with as center draw the arc P'Q' to meet OP in 
Q'; produce PP' into PS'=N.PP', and the chords 
from P to the arc PP' in the same ratio, so as to deter 
mine the arc PS' similar to the arc PP'; draw S'O' 
parallel to P'O to meet PO in 0', and with 0' as 
center draw the arc S'R" to meet OP in R" ; then 
by construction, and similar figures, <POP'=A0, 
arc Q'P'=r' A 0, PQ' = Ar, arc PP'=As, and arc 
R"S' = N arc Q' P' = r' NA 0, PR" = NP Q' =NA r, 
&ygPS'=NAs. When P' approaches P, the arc and 
chord PS' approach the common limit PS=ds on the 
tangent at P (Art. 75); also S'O', which is parallel to 
P'O, approaches a parallel to PO through S, and con- 
sequently the arc R"S', which meets its radii perpendicu- 
larly, approaches coincidence with the line R,S that is 
pependicular to OP at R t . Hence : 

R 1 S=limarcR' , S'=limr'NA0=rdO 
PR, =lim PR"=limNAr=dr 
<POS 1 =SircPS 1 /OP=R 1 S/OP=dO 



72 A PRIMER OF CALCULUS 

Observe that PR 1 S is the ultimate form of the dif- 
ference figure PQ'P', since it is the limit of the similar 
figure PR'S'. 

77. The differential of the ordinate area u=ABLP 
(Fig. 1), is the area (ydx) of the rectangle on LP and PR. 

For Aw=area PLL'P'=y 1 &x, where y l is some 
ordinate between LP and UP' , so that when P' ap- 
proaches P, we shall have lim y±=y; and hence 

(a) du — lim iVA u = lim y x A T A ic = y dx. 

78. The differential of the polar radius area, v=OAP, 
(Fig. 2) is £/*e area of the triangle OPS or of the sector 
OPS,. 

For Av=area OPP'= triangle OPP' plus segment PP f 
or Au== sector OQ'P' minus figure PQ'P'. Also since 
similar areas of Fig. 2 are as N 2 :l, therefore 

N. segment PP' == segment PS" / N,=o when iV= oo , 
:N. figure PQ'P' = figure PR"S'/N, = o when A T =oo . 

Hence dv=lim N. triangle OPP'=lim triangle OPS' = 
triangle OPS, or dv=lim iVsector OQ f P'= sector 0P£, 
since lim OQ'=OP, lim A 7 arc Q'P'= arc P^ . 

(a) dv=j?r 2 dO=^(xdy — ?/(fc) 
For drawing the ordinates iP and KS, 

dy=area OPS=±OP. R 1 S=ir* do 

= i \OK.KS—(LP+KS)LK—OL.LP\ 

=J(xd?/— 1/ da;). 



CONCRETE REP RESENT A TION 



73 



Also, from Fig. 2, 

(h) r 2 =.r 2 +?/ 2 , 0=tan- 1 (y/x), 
which give by differentiation, 

(c) rdr=xdx-\-ydy, r 2 d0 = xdy — ydx. 
From the right triangles PRS, PR l S, we find 

(d) ds 2 =dx 2 + dy 2 =dr 2 + r 2 d0\ 

Also verify the second equation of (d) by squaring 
and adding equations (c). 




79. Observe that as shown in Art, 20, the preceding 
differentials all vary proportionally with dx, for an 
assigned value of x, and each can therefore be expressed 
in the form fx.dx. For example, let the given curve 
be y=x 2 ; then, dy = 2x dx; ds= J (dx 2 -\-dy 2 ) = 
J ( 1 -[-4 x 2 ) dx; du=y dx=x 2 dx; dv=\ (x dy — y dx) —\ x 2 dx, 
etc. Inverse Prin. 1 will then determine s, u, v by 



74 A PRIMER OF CALCULUS 

anti-differentiation as soon as the initial point A is 
assigned; if this be x=o, then 

s =!a-7(l+4z 2 )+ilog(2£+Vl+4z 2 ), 
u=x s /3=xy/3; v=\u. 

80. Definition. The state of change of a variable 
quantity at given values of its variables, is that state in 
which it would change by the value of its differential 
when its variables are changed by the values of their 
differentials. 

E. g., x 2 , at a given value of x, is in a state of change 
in which it would change by 2x dx where x is changed 
by dx; at x=S r x 2 =9, and would become, as it is then 
changing, 15 when #=4, 21 when x=5, and so on; at 
x=5, a 2 =25, and would become 35 when x=Q, 45 
when x=7, and so on. Again, a particle which falls 
16 t 2 feet in t seconds, would, as it is falling at time t, 
fall S2tdt feet in dt seconds, or S2t feet per second. 
So, when we say "that train is running 20 miles an 
hour ' ' we mean to express its state of motion at the 
time of observation, not how far it will run in an hour. 

81. The fundamental properties of the state of 
change of a variable quantity are : 

(a) It is a uniform state; i.e., the changes of the 
quantity vary proportionally with the changes of the 
variables (Art. 20), or more generally the sum of cor- 
responding changes of the quantity and its variables, 
are also corresponding changes of the same, (proven by 

9w 9w .... 

aw=J - dx+ - dy) . 



CONCRETE REPRESEXTA TIOX 75 

(b) The change of value of a quantity in its state of 
change, and its actual change of value, may be made as 
nearly equal as we please by taking the changes of the 
variables small enough — approximate equality between 
very small quantities being taken in the sense that pro- 
portionals to them of sensible magnitude are approxi- 
mately equal, otherwise any two very small quantities 
would be approximately equal, and the stated property 
would be no property. For, let w be the quantity, and 
%, y, its variables; then by definition of differentiation, 
any changes Ax, Ay can be taken so small that for a large 
multiplier N, and the assigned values dx=NAx, 
dy—NAy, to which corresponds div=k, say, we shall 
have NAw and k as nearly equal as we please. Hence 
assigning anew, dx = Ax, dy=Ay, to which corresponds 
dw = k/N (Art. 20), we shall have dw and Aw as 
nearly equal as we please in the sense that their pro- 
portionals k and NAw of sinsible magnitude are so. 

We may say that the differential die, of a function of 
x, y, corresponding to sufficiently small values, dx=Ax, 
dy=Ay, is the principal part of Aw. 

E.g., taking dx = Ax=PQ (Fig. 1) sufficiently 
small, then dy = QT, ds= PT, du = rectangle L Q, 
dv= triangle OPT, are the principal parts of Ay=QP\ 
As=arcPP', Au= area PLL'P', Ay = area OPF. 

82. When a point P is moving upon a fixed curve 
(Fig. 1), the quantities x, y, s, r, 6, u, v, etc., correspond- 
ing to the given position of P have states of change 
characterized by the corresponding changes dx, dy, ds, 
dr, dO, du, dv, etc. These vary proportionally together so 



76 A PRIMER OF CALCULUS 

that the ratio of any two is the change of the first quantity 
■per unit of change of the second. Thus y would change by 
dy, and s by ds, when x changes by dx, and dy/dx, 
ds/dx are the rates of change of y, s, as to x. If P has 
its given position at a corresponding time t, then x, y, s, 
etc , are functions of t, and dx, dy, ds, etc., are differ- 
entials corresponding to any assigned length of time dt. 
Considering the motion of P with respect to change of 
distance, it would move in time dt a distance ds on 
the tangent at P, so that ds/dt is the speed of P at time 
t, and it is tangential in direction. Using a dot over 
a quantity to denote its derivation as to the time, then 
s=ds/dt is a quantity depending upon t, and its 
change per unit of time is s=ds/dt, called the tangen- 
tial acceleration. Considering P as a small particle 
moving in its path in consequence of force acting upon 
it, as in the case of a thrown pebble moving under the 
action of gravity and the resistance of the air, the 
tangential acceleration is not the whole acceleration, 
but only that component of the whole acceleration that 
is in the direction (<£) of motion. The whole accelera- 
tion is by Newton's laws of motion, the time derivative 
of the velocity, of which the speed is simply the magni- 
tude; in other words, variation of direction as well as 
of magnitude, must be taken into account in the differ- 
entiation. (See Art. 85). 

Curvature 

83. The curvature of a curve at any point is its 
change of direction (in radians) per unit length of arc; i.e., the 



CURVATURE 77 

curvature is d<f>/ds tchere cf> is the slope angle (in 
radians) and s is the arc length. 

(a) When the curvature is zero at every point the curve is 
a straight line. \_If d<j>=o identically, then <f>= constant]. 

(b) The curvature of a circle is Ihe same at every point 
and equal to the reciprocal of its radius. 

Let C be the center of a circle of radius a, A its 
lowest point, and P any other point (Fig. 1); then 
<£ = < XMP = < ACP = arc AP / CP = s/a ; thus 
s=a4>, and ds=adcfi, or d$/ds=\/a. 

8J-. The circle of curvature at a point P on a given 
curve, is the tangent circle at that point with the same 
direction and magnitude of curvature as the curve. Its 
radius is therefore R = ds/d<fi, and its center (7=(x^y), 
is distance R from Pin the direction c£-f-~/2. If R i s 
negative this means that the center is actually in the 
opposite direction, since ds/d<$> will be positive or 
negative, according as <£ increases or decreases as s in- 
creases, i.e., according as the curve bends towards the 
direction <f>-{--/2 or <f> — ~/2. Thus equating pro- 
jections of OC and the broken line OPC upon the axes, 
we find 

(a) x= x-\-R cos (<£+- / 2) = x — R sin cf> == x — dy / d<$>. 

(b) y=y-\-Rsm(<l>+x/2)==x+Rcos<t>=x-{-dx/d<t>. 

In rectangular co-ordinates, 

ds=,J(dx* +dy*) = V(l+P 2 ) dx, 

[p = dy/dx= tan <£] . 

d<t>=dp/{i+p 2 ); R=(l+p^/(dp/dx). 



78 A PRIMER OF CALCULUS 

In polar co-ordinates, 

ds = J (r2 d6* +dr2) = J (r2 -f g2) dO, 

[q=dr/d6=r cot ifi] ; 
^ = ^4-^, [triangle OMP, Fig. 1] ; 

(Z0 = c7i// + ^=5( r2 + 2 ^ 2 )^— rdq\/(r*-\-qZ); 
R = (rZ-\-qZ)i/(r*+2q 2 —rdq/de). 

Differentiation of Directed Quantities 

85. The differential of a directed quantity OP (Fig. 2) 
that varies definitely with the time, and whose values add by 
the parallelogram law, is a directed quantity PS, whose com- 
ponent PR^ along OP is the differential of the magnitude of 
OP, and the perpendicular component R 1 S is the product of 
the magnitude of P and its differential change of direction 
(in radians). 

According to the parallelogram law, (true for veloci- 
ties, forces, etc.) OP+PP'=OP', so that A.OP=PP', 
NA. OP=NPP f =PS', and when P' approaches P, 
d.OP=iimPS' = PS, a tangent to s = arc^4P, of 
length ds. Also, the components of PS along and 
perpendicular to OP, are PR 1 =dr, R 1 S=rdO. 

86. The velocity of P is the time derivative of the 
displacement OP. Its magnitude is s, and its direction 
is <£. The acceleration of P is the time derivative of the 
velocity. Its components are therefore s in the direc- 
tion 4>, (tangential) and sd<f>/ dt=s 2 d^/ds=s 2 / R, 
in the direction <$>-{--/ 2 (normal), where R is the 
radius of curvature of the path at P, (Arts. 85, 83). 
If we draw OP in the direction <f>, and in length 



DIFFERENTIATION OF DIRECTED QUANTITIES 79 

s=ds/dt, then the path of P as t varies is called the 
hodogragh of the motion of P; and it appears that the 
acceleration of P is in direction and magnitude, the 
velocity of the corresponding point P in its hodograph. 
If there is no force acting upon the particle P at any 
time, then s = o or s = constant, and d<j>/ds=o, or the 
path is a straight line. No force acting means then 
uniform motion in a straight line. If s=o and 
dcf>/ds= const ant, then the particle is moving uni- 
formly in a circle, and there is an acceleration 
towards the center at every point, of constant magni- 
tude, (speed) 2 /radius. 

In a particle constrained to move in a circle, this is 
the acceleration of the tension along the radius. It 
appears that the normal component of the force on a 
particle is the deflecting component, and the tangential 
component is the speed accelerator. In straight line 
motion, the normal component is zero, since the cur- 
vature is zero. 

Examples III. 

1. In the parabola ay=x 2 find P, and construct PRS 
of Fig. 1, in the cases : x=2a, dx—a; x=a, dx—2a; 
x=o, dx=Sa; x— — a, dx= — 3a; x= — 2a, dx=a. 
With a given value of x, what change is made in PES 
by changing dxf 

2. Construct points (P) and tangents (PS) of the 
semi-cubical parabola ay 2 =x* for x=o, a, 4a. Why is 
there no point corresponding to a negative value of x? 



80 A PRIMER OF CALCULUS 

3. Find one arc of the semi-cubical parabola from 
%=o to any value of a. 

Ans. ,= _{(l + -)_lj 

4. Find P and construct PR l S of Fig. 2 for the equi- 
angular spiral r=ae d , when 0=o, d0=a; = -/6, 
d0=2a; 0=-/3, tl6==a; = ~/2, 60=— a. Show 
that the radius always meets the curve at an angle 
of 45°. 

5. Draw the points and tangents at 0=o, x/4, 
7t/2, 3-/4, -,. for the curves ?^=2acos#, r=a cos20. 

6. Find the curve in which r=a when 6=o, and 
whose polar radius meets the curve at a constant angle 

^=tan —1 c. [rd0/dr = c; r=ae 6//c ~\. 

7. Find the length of the arc of the equi-angular 
spiral of Ex. 6, from 0=o. 

[ds=J(l + c*)dr; s=J(l+c*) (r— a)]. 

8. A point P moves so that its distance (V) from a 
fixed directrix OY is in a constant ratio (e:l) to its dis- 
tance (r) from a fixed focus F; show that a tangent to 
the locus between the directrix and point of contact 
subtends a right angle at the focus. 

[r = ex, dr=edx; take PS = ds, where S is on the 
directrix; then dx= — x, and dr— — exr= — r=PF, 
so that (Art. 76) F=R 1 , the foot of the perpendicular 
from S on FP.~] 

9. A point P moves so that the sum of its distances 



DIFFERENTIA TION OF DIREC TED Q UAXTITIES 8 1 

(r, r') from the fixed focii F, F f , is a constant (2a). 
Show that the tangent at P bisects the angle between 
one focal radius and the other produced. 

r-f-r' — 2a, dr-\-dr r =o; therefore take PR 1 on FP 
produced for dr, whence an equal length PR' \ on PF' 
is dr', and the perpendiculars to r, r' at R 1 , R\ must 
meet in a point S of the tangent at P (Art. 76). 

10. If the point P moves so that the difference of the 
focal radii r, / of Ex. 9 is a constant, show that the 
tangent bisects the angle between the focal radii. 

11. If the focal radii of P as to fixed focii F, F' 
satisfy the condition r-f-2/''— 3a, find a construction for 
the tangent at P; similarly if r — 2r'=Sa. 

12. Find the ordinate areas, and the polar radius 
areas from 0, in the curves ay=x 2 , ay 2 =x 3 , a 2 y=x s . 

13. Find the ordinate area of the circle x 2 -*\-y 2 =a 2 
from x=Oj and of the ellipse x 2 /a 2 -\-y 2 /b 2 =1. 

Ans. i'xj^a*'— x 2 ) +ia 2 sin" 1 -, and b/a times 

the same. The areas of circle a, and ellipse (a, 6) 
are found by putting x=a and multiplying by 4, giving 
-a 2 , -oh. 

14. Find the polar radius area of the circle r=a 
from 0=o. 

15. Find the polar radius area from 0=o of the 
curves of Exs. 1-6. 

17. Find the volume of a hemisphere of radius a, 



82 A PRIMER OF CALCULUS 

between its base and a parallel section at distance x; 
also the convex surface. 

[ The radius of the section at distance x is 
y=J (a 2 — a 2 ), and if V be the required volume 
AF^-^Ai where -y\ is the area of some section 
between the distances x and x-\-Ax, so that \\my 1 =y. 
Thus dV=-y 2 dx=-(a 2 — x*)dx, V=-x(a 2 — %x 2 ) . 
If S be the convex surface, s its arc section by a 
diametric plane perpendicular to the base, then 
dS=2-y ds=2~adx since s is an arc of the circle 
y=J (a 2 — x 2 ) from aj=oy and S=2-ax~\. 

18. Find the moment of the spherical segment of 
Ex. 17 as to its base, and its center of volume. 

[The moment of volume as to a plane is, volume times 
distance from plane to center of volume — distances on 
opposite sides of the plane having opposite signs. The 
moment of a volume equals the sum of the moments of 
its parts. These suffice to determine moment and 
center. Thus, mom F=mom F-f mom A F, and hence 
A . mom V= mom A V=x 1 A V where x r is some dis- 
tance (to the unknown center of volume) between x, 
x', so that lima; 1 =cc, and d. mom V=xdV=momdV, 
considered as concentrated at the distance x to 
which it pertains. Hence (I mom V=-(a 2 x — x s )dx, 

momF=j(2a 2 x 2 — x 4 ), and distance of center of 

3^ 2a 2 x 2 

volume = ( mom V) / V— — . 7r — — . Take x = a to 

4 Sa 2 — x 2 

make F=vol. hemisphere, etc.] 



DIFFERENTIA TION OF DIRECTED Q UANTITIES 83 

19. Find the volume of a right circular cylinder of 
radius x and altitude c and its moment of inertia about 
its axis. 

[AF=2-x 1 Ax r, where x x is between x and x r . 
dV=2~cxdx, V=ttcx 2 . The moment of inertia of a 
volume as to an axis is, the volume into the square of its 
radius of gyration as to the axis. Such radius is between 
the longest and shortest radius to volume. Also the 
moment of inertia of a volume is the sum of the 
moments of inertia of its parts. Thus A . mom-iner. 
V= mom-iner. A V—x\ AT 7 , d. mom-iner. V=x 2 d V— 
mom-iner. d V, considered as concentrated at distance 
x. Moment inertia Y=ttcx± /2) radius of gyration 
=x/V2]. 

20. Find the volume of a cone of altitude x whose 
base area is a 2 when x = l; also find its moment as to a 
plane through its vertex perpendicular to its altitude, 
and the distance of its center of volume from the plane. 

21. Find the moment of inertia, radius of gyration 
about its axis, and the convex surface, of a cone of 
revolution, of altitude x and semi-vertical angle ft. 

I7rx 5 tnii^yl0,xtixnfij. 3, t a 2 sec/? tan /S]. 

22. Show that when x=A. the quantity Jx is chang- 
ing one-fourth as fast as x, and that for small values of 
h, 2-|~i^ is the principal part of ,J(4:-\-h). 

23. A man walks 3 feet per second towards a tower 
80 feet high. If he should continue to approach the 
top as at 60 feet from the base, in what time would he 
reach the top? 



84 A PRIMER OF CALCULUS 

[s 2 =6400+£ 2 , dx—- — Sdt; dt is required when 
x = QO and ds— — 100, and is 55.5 + .. seconds]. 

24. Two men starting together walk in paths at right 
angles, each 3 feet per second; show that one leaves the 
other 3^2 feet per second. 

25. A vessel is anchored in 18 feet of water, and the 
cable passes through a sheave in the bow 6 feet above 
water. If the cable is hauled in 18 feet per minute, 
what is the speed and acceleration of the vessel when 
30 feet of cable are out ? 

[If I = cable out at start, a == horizontal distance 
to anchor, and s, x = same after t minutes, then 
(f— «)2 = (a— a0 2 +2T 2 ; ds=18dt, and when I— s=30, 
we have x=30, a: =32]. 

26. A particle P moves in a plane curve about a 
fixed point 0; find its radial and radial normal com- 
ponents of velocity and acceleration. 

Take an initial axis OX, and let {r,0\ stand for a 
directed quantity whose magnitude is r in the direction 
radians from OX, this symbol in particular standing 
for OP so that 6=:<:XOP, r = length OP. Then by 
Art. 85, velocity 

=d (OP) /dt=d \ r, } /dt= { r, j + j r$, 0+? } =OP 

To differentiate again, we have, 



d { r ,e}/dt={r,e}+{r0,e+l\ 



rO;$+§}—[T»-\*9,0+j]+\r4\0+* 



DIFFERENTIATION OF DIRECTED QUANTITIES 85 
and adding, we find acceleration 

=d(OP)/dt = {r—rO*,6\ + {2r6 + re,6 + Z\ 
The required components are therefore, 

velocity, r = dr/ dt j 

rO=rdO/ dt 

Ati ddr dO* 

acceleration, r — r& 2 =— ^- — r -7—; 
at cit at 2 



*'+*-££ 



t d0 

X * 

dt 



27. If a particle P move about in a plane curve so 
that the radius OP describes equal areas in equal times 
then the whole acceleration is radial, and conversely. [If 
v=ct, then (Art. 78a) dv=r 2 dd = cdt, or r 2 dd/dt=c, 
a constant, and the radial normal acceleration is zero; so 
conversely. The radial acceleration is then the whole 
acceleration and is 

d dr__ dO* _ c 2 ,'fl^} ' 2 
J ~aT~a7 9 "7772 a„2 aa \~L~i aa \ ' 



dtdt dt 2 4r 2 d6 [r 2 dOj 4r ; 
[d_ d^l 1] 

\ AA A A Z I M J 



4r2 U/<9 rf^r 1 rj 

from 1/c/fec/r- 2 c?0. This finds/ as soon as the path 
is known.] 

28. If a planet P move in an ellipse with the sun as 
focus, and the radius OP describe equal areas in equal 
times, show that the force moving the planet is towards 
the sun and varies directly as its mass and inversely as 
the square of its distance. 



86 A PRIMER OF CALCULUS 

[This was deduced by Newton, the premises being 
Kepler's laws obtained by astronomical observations, 
and it led to the law of gravitation. In the ellipse of 
focus 0, major radius a, minor radius 6, excentricity e, 
and direction OX along the major axis toward the 
center, l/r=a(l — ecos 0)/b 2 , and/= — ac 2 /4b 2 r 2 . 
The constant ac 2 /\b 2 is the force per unit mass per 
unit distance, since acceleration = force per unit mass. 

29. If the cubes of the major radii of the orbits of 
any two planets are as the squares of their periodic 
times (Kepler's third law), show that the gravitational 
constant is the same for all planets. 

[The period of one revolution being T, then area of 
orbit — cT=7tab, and the gravitational constant is 
ac 2 /4b 2 = 7z 2 a*/4:T 2 ]. 

30. Find the differential equations of the curve formed 
by a flexible cable with fixed ends and supporting a 
load continuously distributed along the cable. 

[Take a tangent and vertical line through the lowest 
point of the curve (so that the tangent is horizontal) 
for axes of reference ; let P= (x, y) be any point of the 
cable; H the tension at 0, and T the tension at P 
(tensions are tangential because the cable is flexible) ; W 
the load supported by s—slycOP. Then considering 
the equilibrium of arc OP under the forces H, horizontal, 
W vertical, T along the tangent at P, and the differential 
triangle PES, we find H:W:T=PR:RS:P$=dx:dy.ds; 
in particular T 2 =H 2 + W 2 .] 

31. Find the form of arc of a suspension bridge cable 
and the tensions. 



DIFFERENTIA TION OF DIREC TED Q UANTITIES 87 

[Practically, W=cx, and putting H=ca, from 
H:W=dx:dy we find ady=xdx and y 2 =x 2 /2a, a 
parabola. Let (h, k) be one end of the cable; then 
k 2 =h 2 /2a or a = h 2 /2k; T = c J (a 2 + x 2 ), 
=cj(a 2 -\-2ay), =c (a-\-y) approximately, if y is 
small compared with a, i. e., if k is small compared 
with h.~] 

32. Find the tensions and form of arc of a cable with 
uniformly distributed load. 

[Here W=cs, and putting H=ca, then T=cJ(a 2 -\-s 2 ). 
From W:T=dy:ds, we find dy=sds/ ,J(a 2 +s 2 ), 
y= l J(a 2 +s 2 ) — a, T=c(a-f2/). If at one end s=/, 
p=jb, z=/&then £ + &=,,/ (a 2 +£ 2 ), a=(l 2 ^k 2 )/2k, 
— h 2 /2k approximately if k is small compared with 
I so that I is nearly straight. From W:H=dy:dXj 
we find 

dx—ady / s=ady / V (a + 2/) 2 — a 2 / 

a + ?/— V(a+2/) 2 — a 2 ; 

analog — — - — L - i ^ - 

a 



which gives 

n — x 

-( e a +e -a )=a + 2/? 

x 

a catenary. Using the expansion of e a we have approx- 
imately y= x 2 / 2a, a parobola.] 

Curve Tracing 

87. To trace the locus of F(x, y) =o, take a series 
of values of x, and for each value x=a find from the 



88 A PRIMER OF CALCULUS 

equation the corresponding values y==b,b f ... , and plot 
the points (a,b), (n, £/),... The points so plotted on 
each verticle line x=a are the points where the several 
branches of the locus cross that line. When the vertical 
lines are close enough the form and continuation of 
each branch will be shown by its dotted construction. 
This is the primitive method; an improvement consists 
in drawing a short dash at each plotted point in the 
direction of the tangent, when fewer points are neces- 
sary. The tangent is drawn from its slope dy/dx 
which is — F 1 (x,y) /F 2 (x,y) where the numerator and 
denominator are the partial derivatives of F(x, y) as to 
x, y, respectively; viz., since F(x,y) remains zero as x, y 
change continuously, therefore dF(x,y~)=o, or by Art. 
34 (a), F 1 (x,y)dx-\-F 2 (x,y)dy = o. A further im- 
provement is to obtain an accurate idea of the general 
form of the locus from a systematic study of the equa- 
tion, when it will be necessary to plot the locus with 
care only at a few critical points. The methods of such 
study will be considered in detail. 

88. Examine the equation for symmetry as to axes and 
origin. The test of symmetry is that the substitution, 
for the co-ordinates (x,y), of the co-ordinates of the 
symmetric point, in the equation of the locus, must 
leave the equation unaltered. E.g., the locus of 

x ljjl— 1 

is symmetric as to the x-axis because changing (x,y), into 
(x, — ?/), the symmetric point as to the a: axis, leaves 
the equation unaltered. Similarly, this locus is sym- 



CURVE TRACING 89 

metric as to the y-axis because changing (x, y) into 
( — x,y) leaves the equation unaltered; and it is sym- 
metric as to origin, because changing (x, y) into 
( — x, — y) leaves the equation unaltered. In general, 
the substitution, for the co-ordinates x,y, of the co- 
ordinates of the symmetrical point, in any equation, 
gives the equation of the symmetric locus ; e.g., the 
locii of y 2 = Sx + 12 and y 2 = — 8x + 12 found by 
replacing (x,y) in the first by ( — x,y) are symmetric 
locii as to the ?/-axis since if any point (a,b) satisfies 
the first, then ( — a, 6) satisfies the seeond. 

89. Examine the equation for limits of real value of 
x and y. If y is imaginary when x lies between a and 6 
then no part of the locus lies between the vertical lines 
x=a and x=b; for although the imaginary value of y 
is in such case an algebraic solution of the equation, 
and (x, y)) is a point of the locus in an algebraic sense, 
yet no point in the plane of representation corresponds 
to it. E.g.. in x 2 /a 2 -\-y 2 /b 2 = l, where a,b are 
real numbers, if x 2 ><2 2 then y is imaginary, and if 
2/ 2 >6 2 then x is imaginary; hence the locus lies be- 
tween the vertical lines x=±a, and the horizontal 
lines y — ±b. 

91. Directions to and at infinity. The direction whose 
slope is the limiting value of y/x=t&nQ as the point 
(x, y) approaches infinity on a distant branch is the 
direction to infinity of that branch. The direction at 
infinity of the branch is the direction given by the limit 
of the slope c?2//cfa;=tan<£ at the distant point (x, y) 
on the branch. This is identical with the direction to 



90 A PRIMER OF CALCULUS 

infinity; for limy/x=limdy/dx by the theory of 
indeterminate forms, when both x and y approach 
infinity, and when limy/x=o or oo in consequence 
of y or x approaching a finite limit then also 
limdy/dx=o or go. E.g., if for a finite value x we 
have y = cc , then x' being near to x, y' will be finite, and 
Ay=y' — y=cc , so that dy=cc when dx is infinite, 
and dy/dx=oo=y/x. It appears that a distant 
branch with a limiting direction is very nearly a straight 
line of slope \\m_dy / dx = \\my /x. A spiral winding 
indefinitely around the origin and extending indefinitely 
outward is an example of a distinct branch with no 
direction to or at infinity. 

92. Asymptotic Line. When all the points of a 
distinct branch approach more and more nearly coinci- 
dence with the distant points of a given straight line, 
such line is the asymptote of the branch. To have an 
asymptote, it is evident that the branch must have a 
direction to infinity; also, the tangent to the branch at 
a point approaching infinity must have this asymptote 
for its limiting position, so that the asymptote is a line 
tangent to the branch at infinity. If (x, y) be a point on 
the branch approaching infinity, then excepting vertical 
branches for which limy/x—co , we shall have 
limy/x=m, the slope of the asymptote, and the 
ordinate of the asymptote that passes through the 
point (x, y) on the branch will be mx-\-b, where b is 
^-intercept of the asymptote. The condition that the 
branch approaches coincidence with the asymptotic 
line is that the difference of their ordinates to the same 



CURVE TRACING 91 

abscissa, or (mx-\-b) — y, opproaches zero when x 
approaches infinity. Thus 6=lim (3/ — rax). Denoting 
y — mx by q, we have then to put y=mx-\-q in the 
equation of the locus and in the resulting equation for 
q in terms of x find the limiting value of q as x 
approaches infinity, under the condition that q is to be 
finite. If the equation between q and x is algebraic, 
we divide it by the highest power of x and find the 
limiting equation by putting x=oo . The value of q 
from this equation is the required ^/-intercept of the 
asymptote. 

83. For example find the directions to infinity and 
the asymptotes of xy 2 — x 3 — 2ay 2 -j-a 3 =o. Divide 
this by x 3 and put z=oo ; if y be supposed finite we 
obtain = 0, which finds no finite value of y when 
a; =00 ; if y/x be supposed finite and equal to m, we 
find m 2 — l = o, which gives two directions to infinity 
of slope 1 and — 1 respectively. Divide the equation 
by y 2 and put y = cc ; if x be supposed finite, we find 
x — 2a =0. This is therefore a vertical direction to 
infinity whose asymptotic line is x = 2a. To find the 
asymptotes of slope m=l or — 1, put y=mx-\-q in 
the given equation, and it becomes, since m 2 =l, 
2 (mq — a) x 2 -{-(q 2 — 2amq)x — 2aq 2 -\-a z =o; dividing 
this by x 2 , we see that if q remains finite as x increases 
indefinitely, we must have in the limit, mq — a=o or 
q = a/m=7na. Thus y=m(x-\-a) = ± (x-\-a) is the 
equation of the asymptote of slope m=±l. 

(a) Show that, the terms of highest degree in the equation 
of a locus, when equated to zero, give the equation of the lines 



92 A PRIMER OF CALCULUS 

to infinity through the origin; viz., in the above example, 
xy 2 — £ 3 =o, are such lines, etc. 

94. Examine the equation for regions of rising and 
falling branches (from left to right) and resulting crests 
and hollows. 

In other words, note from the equation where y in- 
creases, where y decreases, and where it is at a maximum 
or minimum value. If the equation does not show 
this readily, it is determined by the values of (x, y) that 
make dy/dx positive in the first case, negative in the 
second case, and where dy/dx is changing sign in the 
other cases. 

95. Examine the equation for regions of concavity 
upward, concavity downward, and consequent points of 
inflection, 

In other words, note where the tangent turns counter- 
clockwise as its point of contact advances to the right 
(which is shown by the slope p=dy/ dx increasing or 
by dp/dx positive) where the tangent turns clockwise 
(which is shown by the slope p=dy/dx decreasing or 
by dp/dx negative) and where the tangent is changing 
direction of turning (which is shown by dp/dx chang- 
ing sign.) 

96. Examine the equation for multiple points, and trace 
the locus in Ihe neighborhood of a multiple point. 

A multiple point is a point where two or more 
branches of the locus intersect; it is a double or triple 
point, etc., according to the number of branches. 



CURVE TRACING 93 

At such a point p=dy/dx= — F 1 (x,y) /F 2 (x,y), 
(Art. 87) must be correspondingly multiple valued, 
which can only be (excluding discontinuity) when this 
fraction is o/o for the point (x, y). Thus a multiple 
point must be a solution of the simultaneous equations 
F(z,y)=o, F 1 (x,y)=o, F 2 (x,y)=o. For such a 
point we have by the theory of indeterminate forms, 
p= — dF 1 (x,y)/dF 2 (x ) y) which becomes, after re- 
placing dy/dx by p, and x, y by their values at the 
multiple point, a quadratic for p. If this quadratic 
is determinate the point is a double point; and the 
branches intersect, or touch, or are imaginary, accord- 
ing as p has two different or eqnal real values or two 
imaginary values. In the latter case the multiple point 
is an isolated point of the locus with no real point next 
to it. In general, if (x 1J y x ) is the double point, we 
put x=x 1 -\-h, y=y } -\-k in the equation of the locus, 
and trace the locus for small negative and positive 
values of h. This is, in effect, transforming the axes 
to parallel axes through the multiple point, with h, k 
as co-ordinates to the new axes. It is easily shown 
that the terms of lowest degree in the resulting equation 
for (h, k) when put equal to zero, will be the equation 
of the tangents at the multiple point, which is now 
made the origin; and this equation determines at once 
by its degree, the order of the multiple point. If m be 
the slope of one of these tangents, we trace the branch 
to this tangent by putting k=mh-\-q in the equation 
of the locus, aud determine from the resulting equation 
in q, h, the principle part of q for small positive and 
negative values of h, and thence the position of the 



94 A PRIMER OF CALCULUS 

corresponding point (h, Jc) above or below the tangent, 
according as q is positive or negative. 

97. Trace the locii of the following equations, taking 
convenient lengths for the constants. 

2ay=x 2 — 2ax-\-ab; 
3a 2 y=x* — 3aa; 2 +3 (a 3 ±6 3 ) x+a 3 c (b>o) 



X 2 „2 



a 2 b 2 






a 2 2/ 2 =a 2 x 4 — x 6 . 

?/ 2 ==a; 8 /(2a — x), the cissoicl; if OP intersect the ver- 
tical line x=2a in Q, and the circle on the abscissa 
2a as diameter in R, show that OR = PQ. 

ay 2 =x s ; ay 2 = (x—ay(x—b) (a>, <,=&). 

sd-?(-yi==a§, the hypocycloid described by the rolling 
of a circle of radius a/ 4 inside a circle of radius 
OJ.=a, the tracing point being on the circumference of 
the rolling circle and on the axes when in contact with 
the fixed circle. 

y=Sa s / (x 2 +4a 2 ), the witch; draw a circle with 
vertical diameter OA— 2a; draw a line from to 
meet the circle in R and the tangent at A in Q, when 
the abscissa of Q and ordinate of R are the (x, y) of a 
corresponding point P of the witch. 

x s -\- yz = 6ax 2 ; x z -\- y z =Zaxy ; 
x 3 -\-y* = a s ; x* — 2a 2 y = ay z . 

(x 2 -{-y 2 ) 2 =a 2 (x 2 — y 2 ), the lemniscate, if r, r' be 
radii to P from focii F, F' on the z-axis such that 



CURVE TRACING 95 

F'0=0F=a/ l J2, then r/=a 2 /2; with polar 
radius r from 0, r 2 = a 2 cos 20. 

a - ~~ a; 

2/= — (e a + e a )=a cosh -, the catenary. 
a a 

98. The same methods may be employed in tracing 
the locii of polar equations. In looking for symmetry 
the symmetric of (r, 0) has several forms that must be 
tried separately: e.g., the symmetric points of (r, 0) as 
to OX are (r, — 0), ( — r,- — 0), etc. A direction to 
infinity is a value of that makes r=ab , and the cor- 
responding asymptote is found from the limit of the 
polar subtangent OM'=r 2 dO/dr, whose direction of 
measurement is — r/2. A direction that makes 
r=o is the direction of a tangent at the origin, since 
that is the limiting direction of the chord OP as P 
approaches 0. The locus recedes or approaches the 
origin as increases if r 2 is increasing or decreasing, 
and it is concave towards or from the origin according 
as t&nif/=rd6y'dr increases or decreases with 0. The 

following equations are given to trace : 

6 

r=aO, the spiral of Archimedes; r=ae c , the equi- 
angular spiral. 

r=2asin0, r = 2acos0, r=a sec20, r = a/0. 

2 3 

r=asec— ; r=a sin 2$, r=a cos 20; r—a sin ^ . 

r 2 =a 2 cos20/ r=a(l — cos0), the cardioid. 
r=a-\-b esc 0/ r=a (sec20-f-tan 20)/ 
r=a 2 esc # 2 +& 2 sec 2 . 
r=acos0-|-6 sin/9/ r = a cos26-\-b sin20. 



96 A PRIMER OF CALCULUS 

Envelopes 

99. Let three variables x, y, t be always conrfected 
by a given equation F(x, y, t)=o; then to a given point 
(x,y) corresponds one or more numbers (the numbers 
of P) which are the solutions of the given equation for 
t in terms of the given values of {x,y). We assume 
every point of the plane to be so numbered by this 
equation. The equation F(x,y,t)—o may be the 
equation of any locus we please in the plane by select- 
ing for t a proper functional value t=f(x,y). In other 
words, consider any given locus in the plane, and select 
from point to point of that locus one of the numbers of 
each point so that this number varies continuously 
with the position of the point; then the continuous 
assemblage of such numbers form a definite function 
t—f(x, y). It is obvious, on account of the multiplicity 
of the numbers of each point, that it may be possible 
to find different functional values of t such that for each, 
F(x, y,t)=o shall be the equation of the same locus. 
Since dF(x,y, t)=o on such curve, we have (using the 
notation of Art. 34 for partial derivations as to the 
first, second, and third variables) 

(a) F 1 (x, y, t)dx+F 2 (x, y, f)dy + F s (x, y, f)dt=o. 

This is an equation for the slope dy/dx at the point 
(x, y) on the given curve, remembering that dt is of the 
form Ldx-\-Mdy depending upon the given locus. 

100. The n-curve. The locus of all points having 
the same number n is the ?i-curve. Its equation is 
F(x,y,n)=o. Thus the o-curve is F(x,y,o)=o, the 



ENVELOPES 97 

i-curve is F(x,y, l)=o, etc. The slope of the ?i-curve 
is given by t=n, dt — o in 99a and is 

(a) F x (x, y, n) dx + F 2 (x, y, n) dy=o. 

101. The self -intersections. The points (x ) y) that 
simultaneously satisfy F(x,y,n')=o F(cc,y,n) = o may 
be called the n'. n points, because they are each points 
of number n' and n; the} 7- are the intersections of the 
?i'-locus with the w-locus. The limiting positions of 
these intersections as n' is taken nearer and nearer and 
to its limit n are the n.n points, or self -inter sections of 
the n-locus. To find these n.n points we must replace 
the n'-locus by another that always intersects the ?i-locus 
in the n' . n points and only those, and that does not 
reduce to the n -locus itself when we put n'=n. This 
locus is given by 

[F(x, y, n')—F(x, y, n) ] / (n'-~ n) =o. 

since any point (x. y) that satisfies this equation and 
F(x } y,n)=o, will also satisfy Z (x, y J n')=o and so be 
an n'. n point, and conversely every n'. n point satisfies 
the above equation. Taking n as the original value 
and n' as the new value of t, so that the equation is, 
A t F(x,y,t) / At = o, we see that its limit is F B (x,y,ri)=o. 
Hence 

(a) The self-intersections of an n-locus are its intersections 
with the locus of the partial derivative of its equation as to 'its 
number {regarded as the original value of the variable t) ; 
i.e., the n.n points are the values of (x, y) that simul- 
taneously satisfy F(x, y, n) =o, F z (x, y, n)=o. 

102. By solving the preceding simultaneous equa- 



98 A PRIMER OF CALCULUS 

tions for (x, y) we find the co-ordinates of self-intersec- 
tion of the n-locus each in terms of the number n of 
that locus; then by giving n all values, we find the 
assemblage or locus of all n.n points for all values of n, 
or the locus of self intexsections. By eliminating n between 
the above simultaneous equations (by solving one for 
n and substituting its value in the other) we evidently 
obtain the equation of the locus of self intersections in 
terms of (x.y) alone. We may select the variable 
function t=f(x, y), so that F(x,y,t)=o is the equation 
of the locus of self-intersections, viz., f(x,y) is a solu- 
tion of F s (x, y,t)=o for t in terms of x, y. The slope 
of the locus of self-intersections is then given by 99(a) 
which reduces since F s (x,y,t)=o to 

(a) F 1 (.r, y, f) dx+F 2 (x, y, t) dy=o. 

103. The multiple points of an n-locus are points of 
self-intersection of that locus. For at a multiple point 
(x, y) of the w-locus, F(x,y, ri)—o, we have also 
F 1 (x,y,n)=o, F 2 (x,y,ri)=o (to make dy/dx=o/o). 
Thus, substituting t=n in 99(a) which is true for all 
values of (x, y) of number t, even when dt is not zero, 
we find that F s (x,y,n)=o; i.e., by 101a, the multiple 
point (x, y) on the w-locus is a point of self-intersection. 
We divide the locus of self-intersections into that of 
ordinary points and that of multiple points of the 
n-locus. 

104. The locus of ordinary self-intersections is met 
tangentially at each point by the n-locus on which that point 
is a self -inter secton. For the slope of the locus of self- 
intersections at such point (x,y) whose number is n, is 



ENVELOPES 99 

found by making t=n in 102a, and since the differ- 
ential co-efficients are definite non-zero values (because 
the point is an ordinary point on the n-locus) therefore 
this slope is the same as the slope of the n-locus, given 
by 100a; i.e., the two locii meet tangentially. This 
result does not hold on the multiple point locus, since 
then F 1 (Xjy J n)=o F 2 (x,y,ri)=o; and in general 
dy/dx—o/o signifies that the value oidy/dx at such 
limiting point depends upon the manner of approach 
of (x, y) to their limiting values, and is otherwise 
absolutely indeterminate. Now on the n-curve, we are 
to find the limit of — F 1 (x, y, n) /F 2 (x, y, n) as (x, y) 
approaches its limit on the locus F{x, y, n) =o; and on 
the multiple point locus we are to find the limit of 
— F 1 (x,y,t) / F 2 (x,y,t) as x, y, t approach their limiting 
values wherein t is a variable approahing n and con- 
ditioned by F s (x,y,t)=o. These are certainly differ- 
ent methods ot approach and give in general different 
limiting values for dy/dx. E.g., on the n-curve 
(y — n) 2 = (x — a) 3 the point x=a, y=n is a multiple 
point whose locus, as n varies through all values, is the 
vertical line x=a. This is the only locus of self- 
intersections, as may be shown by eliminating n be- 
tween this equation and then-derivative, y — n=o; and 
its slope is dy/dx=cc at every point. On the contrary, 
the n-curve has two branches y — n=±(x — a)* that 
meet to form the multiple point at x=a, y=n, and 
on these branches dy /dx=±\(x — a)* whose limit 
when x approaches a is zero, i.e., the slope of every 
n-curve is zero at its multiple point, and it therefore 
meets the multiple point locus everywhere at right 
L.ofC. 



100 A PRIMER OF CALCULUS 

angles — quite the reverse of tangential meeting. In 
general any definite motion of a curve with a multiple 
point, as a lemniscate which is a figure 8, generates a 
system of n-curves, in which n may be taken as the 
time at which the generating curve is an w-curve; and 
such motion can be so determined that the locus of the 
multiple point shall meet the w-locus at any angles we 
please, constant or varying with the locus. 

105. Jf a given locus is met tangentially at every point 
by an n-locus through that point, then it is a locus of self - 
intersections. For, take the equation of the given locus 
as F(x,y,t)—o where the variation of t with (x,y) is 
determined by the condition that t—n at the point of 
tangency (x, y) of an n-locus. The condition of tangential 
meeting at (x,y) is then that the slope given by 99a 
when t = n is identical with the slope given by 100a. 
Thus, making t—n and subtracting, remembering that 
t is a variable so that dt=o for any continuous series of 
values of (x, y) on the given locus is inadmissible, we 
find F 3 (x,y,n) — o i.e., any point (x,y) of the given 
locus is a self-intersection. The complete locus that 
satisfies the above condition of tangency or envelop- 
ment by the ?i-curves will be called the envelope of the 
system of ^-curves. This envelope will not in general 
include the multiple point locus and will be simply the 
locus of ordinary self-intersections. 

106. Let F(x,y,f)—o be the equation connecting 
the volume x, pressure y, and temperature t of a unit 
mass of gas; then the n-curves of this equation are the 
so-called isothermal lines of the gas of temperature n. 



ENVELOPES 101 

/ 

For the so-called perfect gas xy—at, and the isotherm als 
are hyperbolas. An intersection of two isothermals of 
different temperatures implies an unstable condition of 
the gas, and is in general impossible. 

107. Let F(x,y,z)—o be the equation of a surface 
in which XOY is a horizontal plane, and z is the height 
at the point x,y; then the n-curve F(x,y,n)=o is the 
contour line, on the plane XOY, of points on the surface 
of altitude n. In contour maps, we have also no inter- 
section of contour lines of different altitudes, because to 
each point (x, y) corresponds only one altitude number. 

108. Find the envelope of the following systems of 
curves, m, p, q, t, etc., denoting variable parameters. 
For straight line systems draw also a sufficient number 
of lines of each system to show the envelope graphically. 

(a) y=mx-\-a/m; y 2 =4:ax 

(b) (y— mx)2=a 2 m 2 ±5 2 ; x 2 /a 2 ±y 2 /b 2 = l 

(c) x cos t-\-y shit — a—o; x 2 -\-y 2 =a 2 

(d) If points Q, R, move uniformly along straight 
lines OA, OB, show that QR envelopes a parabola. 

[x/t+y/(at + b)=l; (ax+ y y+2b(ax—y)+b*==o-\ 

(e) Find the envelope of a line of constant length 
( a) moving with its ends in the axes. \_x /p -f- y / q = 1 , 
p2_|_g2_ a 2 ? treat p, q as functions of t, then on the 
envelope xdp/p 2 -\-ydq/q 2 =o, pdp-\-qdq—o, which 
gives x/p z =y/q z by eliminating dp, dq. To elimin- 
nate p, q denote for the moment the common value of 
the members by r; then substituting in preceding 
equation gives r—1/ a 2 , etc. Ans. X3-{-y%=a%'] 



102 A PRIMER. OF CALCULUS 

(f) Particles are started from the origin with equal 
speeds in varying directions 0, in a vertical plane; find 
the envelope of their paths. 

[x = at cos 0, y=at sintf — gt 2 /2, and the path is 
y = xtsm0— gx 2 /2a 2 cos0 2 . Ans. y=a 2 /2g— gx 2 /2a\ 

(g) Find the envelope of the variable ellipse 
x 2 / p 2 -\-y 2 /q 2 =l of constant area rca 2 (pq=a 2 ); 
also the one of fixed director circle (p 2 -\-q 2 =a 2 ) ; 
also the one in which p-\-q = a. 

Ans. 4x 2 y 2 =a 2 ; (x±y) 2 =a 2 ; xi+y%=ai. 

(h) Find the envelope of the normal to the parabola 
y 2 =4ax. [y=m(x — 2a) — am s ; 27ay 2 =4(x — 2a) 3 ] 

(i) Show that the self-intersection of the normal of a 
given curve at x, y is the center of curvature of the 
curve at (x, y). 

[y — y= — p(x — x) /p where x is the variable para- 
meter, y a function of x given by the equation of the 
curve, and p — dy/dx. Ans. n=x — p(l-\-p 2 )/q, 
J='y-{-CL-\-p 2 )/q, where q—dp/dx, and these are 
the co-ordinates of the center of curvature (Art. 84.)] 

(j ) Show that the normal of a curve regarded as a 
rigid line terminating in the corresponding point (x, y) 
of the curve rolls on its envelope without slipping. 
[x=z — .Rsin<£ gives dx—dx — Rcos<j>d<f> — sin 4>dR 
= — sin <f> dR, since R cos <j> d <f> = ds cos 4> = dx. But if s be 

the arc of the envelope. dx= c?s cos (<£-}--) = — c?ssin</>. 

Hence ds=dR. and hence As=Ai?, or increase of dis- 
tance R on the normal between x, y and point of con- 
tact x, y equals arc rolled over.] 



INTEGRATION 103 

CHAPTER IV 

Integration 

109. Variation. To give a variable a', a series of 
values one after another, determines a variation of x. By 
any variation of x, a variation of Ax is also determined, 
and any function of x and Ax also varies through the 
same number of values as Ax. E.g., x=l, 3, 7, 10, 
is a variation of x from 1 to 10, in which A.t=2, 
4, 3 where successively Ax=2 corresponds to x=l, 
Ax=4 to x=S, A.i-=3 to x=7; we have also 
xAx=2, 12, 21; x*Ax=2, 36, 147; (x^ -\-x)Ax=4, 
48, 168; (z 2 + zsinAaOAz=2+sin2, 36+l2sin4, 
147 + 21 sin 3; and so on. Another variation from 1 
to 10 is x=l, 2, 4, 5, 7, 9, 10; in which Ajc=1, 2, 1, 
2, 2, 1; zAz=l, 4, 4, 10, 14, 9; z**x=l, 8, 16, 50, 
98, 81; (x*+x)fix*=2, 12, 20, 60, 11, 90; etc. Another 
variation x = l, 2, 3, 4, 5, 6, 7, 8, 9, 10, in which 
Aoj = 1, always, is on that account called uniform 
variation. The general variation of n changes from a 
to u will be denoted by x—a,b,c,... l,u, in which 
Ax = b — a, c — b, ... u — I. 

The final value x=u y may be considered either as a 
constant or a variable, — in the latter case u is another 
symbol for an original value of x, regarded as reached 
by variation from the initial value a ; and it may be 
considered as used merely to prevent confusion between 
the final value and the intermediate values. 



104 A PRIMER OF CALCULUS 

110. Summation. The symbol ^/(x, Ax)= "the 
sum from a to u of/(x, Ax) 5 ' stands for the sum of the 
values of f(x, Ax) in a variation of x from a to u. E. g., 
when x=l, 3, 7, 10, SfflAx=35, 2jV Ax=185, 
^i°C^ 2 + ^) Ax.=220. When x=l, 2, 4, 5, 7, 9, 10, the 
same sums are 42, 254, 296, respectively. When 
x==l, 2, 3, 4, 5, 6, 7, 8, 9, 10, the same sums are 45, 
285, 330, respectively. A sum will therefore in general 
depend upon the variation of x between its assigned 
limits. If it is a variable in the above sum, such sum 
is what is called an imperfect function of u, that is, it 
depends upon, but is not determined by the value of u • 
it requires the method of variation of the variable x 
from its initial value a to its final value u to be also 
assigned. This imperfect function has, however, a 
definite difference f(u, Aw), since changing u to u-{-Au 
changes the sum by the term f(u, Aw) ; i. e. , /(x, Ax) is 
the difference of the sum for any final value of x. It is 
in general an imperfect difference i.e., not the difference 
of a function. 

111. Theorem 1. The sum from a to u of the differ- 
ence of a function is equal to the change from a to u of the 

\ u 
function. In symbols, % a A </>x= 4> x =<f>u — 4> a - 

For the successive values of A<£x in any variation 
x=a,b,c, ... l,u, are <j>b—<f>a, <f>c — <j>b, ... <$>u — tj>l } 
and adding them, the intermediate values <f>b, — <f>b, 
<j>c, — cj>c, ... 4>l, — (f)l, all cancel, leaving only <$>u — <j>a 
for the sum. This result may also be stated as follows: 
The sum of the successive changes of value of a function is 



INTEGRATION 105 

its tatal change of value. As an illustration verify that 



10 



Sj°A.a;2==Sj (2xAx + Ai" 2 ) = I £ 2 = 99, for each of 

the variations of Art. 109. 

(a) It appears that 2^' &<f>x is a definite function of 
u, i. e.j independent of the variation from a to u, and 
that it is "a function whose difference is A<£it"; *. «., 
2 = A-i. 

112. As exercises, show that for any given variation 
x=a, b,c, ... g, h, i, ... I, u, we have: 

(a) l u a [</f(x,Ax-)+c"f 1 (x ! \x)-] 

= C 'S^/(.r,A.r)+c"S:/ 1 (.r,A.r)- 
In words, the characteristics of summation for a given 
variation, is distributive over a sum and commutative with 
a constant factor. 

(b) %y(x, A.r) = 2 ( V(.r,A.f)+S^/fe Ax). 

(c) l tt J(x, Ax) = s;/(z,-Ai) 

113. Continuous variation. Let there be an unending 
number of successive variations from a to u, of greater 
and greater number of changes, and formed according 
to some definite law. In the variation of n changes, 
every value of Az will be smaller* than some number 
h n ; h n cannot be smaller than (u — a)/n, the value of 

*We use the terms "larger ' and "smaller'- with, reference to com- 
parative magnitudes, which are positive numerical values. The magni- 
tude of any number is the positive square root of its square, if it is a rea 
number, and in general, it is the positive square root of the sum of the 
squares of the real components of the number. 



106 A PRIMER OF CALCULUS 

Ax when the variation is uniform, but it is otherwise 
any number we please according to the variation. If 
we form an unending series of variations in such a way 
that the superior limit h n , of every Ax in the variation 
of n changes, approaches zero as a limit as n approaches 
infinity, then such a series will be called an approach to 
a corresponding continuous variation from a to u. In 
other words, in the variation of n changes, when n is 
very large, the values of Ax will each be very small, so 
that the variation is approximately continuous. There 
are many kinds of continuous variation from a to u 
according to the defining series of variations of greater 
and greater number of changes and smaller and smaller 
values for each change. Uniform continuous variation 
is the limit of a series of uniform variations, in which 
h n = (u — a) /in. In a non-uniform variation, however 
nearly continuous, some changes may be very many 
times larger than other changes, although every change 
may be very small. 

114. Continuous summation — Integration. The limit 
of the sum of a difference f(x, Ax), as the variation of x 
from its initial to its final value approaches continuous 
variation, is a continuous sum, or integral. For example, 
we will find the values of the integrals, limS^xAx, 
lim 3^x 2 Ax, lim S^xAx 2 for uniform continuous varia- 
tion. The uniform variation of n changes from a to 
u is x = a, a-\-h, a-\-2h, ... a-\-(n — l)h,u, where 
nh=u — a —n Ax. With this variation the above sums 
are, since l+2 + ...+n — l=n(n — 1)/2, and 
I2_j_2 2 + ... + (n — l)>=n(n— -1) (2n—l)/6, 



INTEGRATION 107 

the first =h[na + n(n — l)h/2~] 

= (u 2 —a 2 )/2-h(u—a)/2', 

the second = h[na 2 -\-n(n— l)ah-\-n(n — 1) (2n—l) h 2 / 6~] 
=s ( u 3-a*)/3-h(u 2 -a 2 )/2+h 2 (u-a)/6; 

the third —h times the first. 

Hence the limits of these sums for n = oo or h=o 
are (u 2 — a 2 )/2, (u z — a 3 )/3, and o, respectively. 

115. We shall show that under certain conditions 
an integral or continuous sum exists in which dif- 
ferent methods of approach to continuous variation 
have no effect upon its value; and we shall assign a 
notation that embraces in compact form the fundamental 
facts and ideas of such limiting sum, and determine, 
by a fundamental theorem, shorter methods for evalu- 
ating integrals than the full process, which is compli- 
cated even for uniform variation, (See Art. 114). 

We consider in the first place, only sums 2 "/(a:, Ax) 
in which the proportional difference Nf(x, Ax) ap- 
proaches a definite differential in terms of x, dx when as 
usual, N approaches infinity and Ax approaches zero so 
that iVAx approaches any assigned value dx. 

There are, in fact, no general methods for determin- 
ing whether an integral exists or not when no such 
differential exists. As in Arts. 20, 21 the differential, 
lim Nf(x, Ax) is of the form <f>' x . dx when x is a real 
variable, and may be so whether x is real or imaginary. 
E.g., lim Nx 2 \x=x 2 dx, lim N(x 2 -f- x sin Ax) Ax = x 2 dx, 
limiVA(x 3 / /3)==x 2 dx. As in these three examples, 
so in general, many different differences will lead to the 



108 A PRIMER OF CALCULUS 

same differential, and some of those differences must 
be perfect when the differential is perfect. 

116. Lemma. An integral over any range of variation 
is zero when the differential pertaining to the integral is 
identically zero. In symbols, lim 3^/(x, Ax)=o, ivhen 
lim Nf(x, Ax)=o t identically. For let x be that value 
in any variation of n changes which corresponds to the 
largest term /(x, Ax) among the n terms of this type; 
then the sum of the n terms is certainly not larger than 
nf(x, Ax), (the magnitude of a sum cannot exceed the 
sum of the magnitudes of its terms); but by hypothesis, 
and for the special case N=n, every product of the 
type nf(x, Ax) approaches zero when n approaches in- 
finity and Ax, zero; and hence the given sum, that is 
always not larger than one of these products, must 
approach zero. 

117. Theorem 2. Integrals with identical differentials are 
also identical, for any same method of approach to continuous 
variation. In symbols, if lim Nf(x, Ax) = lim N<f> (x, Ax), 
then lim^/(x,Ax)=lim2^(x,Ax). 

For the difference of these integrals is the limit of 
the difference of the corresponding sums, which is 
lim ^[/(x,Ax)— <j>(x, Ax)], by Art. 112a; and this 
integral is zero by the preceding lemma, since by 
hypothesis lim N [/(x, Ax) — <f> (x, Ax)] =o identically. 
While the two integrals are identical for any same patu 
of integration from a to u, i. e., for the same method of 
approach to continuous variation from a to u, yet each 
may change value with change of path of integration. 



INTEGRATION 109 

118. Notation. Since the value of the integral 
depends only upon the corresponding differential and 
the path of integration, whether the difference of the 
sum has one or another of the many values whose 
proportionals approach the given differential, therefore 
the integral is named most definitely as "the integral of 
the differential over the given path ." Since we can write 
lim % u J(x, Az)=lim t^N- 1 . lim Nf(x, Ax) by multi- 
plying by N~ 1 N=1 at each stage of the approach, we 

therefore find limS iV"~ 1 = I say, as the characteris- 

tic of integration of the differential from a to u\ and if 
lim Nf(x,'Ax) = lim N<f> (x, Ax) = . . . =<f>'xdx, the identi- 
cal integrals lim ^f(x, Ax) =lim 2 u a 4> (x, Ax) = . . . , are 

each expressed by i cf>'xdx== "integral from a to u of 
tfxdx." 

119. Theorem 3. The integral of the differential of a 
fanction is independent of the path of variation betiveen given 
limits, and is equal to the total change of value of the function 
between the limits. 



d<f>x = \ 4>x. 



For since lim NA<f>x=d<f>x, therefore by the notation 
established by Th. 2, and by Th. 1, 

d<f>x= lim 2 1 A <£ x = lim (4> u — 4> a ) = <t>u — <£&• 
From this result follows 

d<fcx="a function whose differential is" 

a 



110 A PRIMER OF CALCULUS 

dcf>u = d~ 1 . dcj>u, and that vanishes when u=a] i.e., 
f=d-K 

Since d—lim iVA therefore, formally, d _1 =limA- 1 i\- 1 
=lim$N- 1 = C, from ^=A~\ This result shows that 

the formal relations ofd, I , are consistent, i.e., in ac- 
cord with established facts. 

120. Inverse Pr in. 1. If dcf>x=dij/x when x varies, 
then is cf>x — \f/x a constant for variations of x. 
For Irom the given identity and Th. 2 



dcf>x= I d\p 

a J a 



i. e., by Th. 3, <f>u — <pa=\j/u — \j/a, or <£tt — i//u=<£a— ■\pa > 
a constant for variations of x=u. 

121. The preceding results extend to variations, 
sums and integrals, in any number of variables. Thus 
a variation of (x,y) will be a series of sumultaneous 
changes of (x, y) from given initial values {a, 6) to any 
final values (it, v) ; such variation determins a series of 
values of (Ax, Ay) and a series of values of any function 
f(x, Ax, y,Ay). There is no change in any of the pre- 
ceding theorems and proofs except the slight changes 
consequent upon the introduction of the additional 
letters required for the values, changes of values and 
limits of the additional variables. The differentials in 
real variables x,y, etc., are of the forms 

/' x . dx, f t (x, y) dx+f 2 (x, y) dy, etc. 



INTEGRATION 111 

The differential f'xdx where fx is a continuous 
function of x is always a perfect differential dfx. E. g., 
iff'x is real, then draw the curve y=f'x, when the ordi- 
nate area, fx, of that curve from x=a to any final value 
ot x, is a function of x whose differential has been shown 
to be dfx=ydx—fxdx, (Art. 77). The differential 
in more than one real variable is not, however, in 
general perfect, since this requires that the differential 
co-efficients be partial derivatives of a given function of 
the variables. The differentials in two real variables 
x, y include differentials in an imaginary variable 
z=x-\-yJ — 1. When the differentials are imperfect 
then their integrals between given limits (a, b), (u,v) 
depend upon the manner of continuous variation, 
which is called the path of integration. When the dif- 
ferentials are perfect their integrals are independent of 
the paths of integration and functions of the final values 
of the variables (the initial values being given con- 
stants). A path of integration is determined when the 
corresponding values of the variables are determined in 
terms of one real variable, since this reduces the differ- 
erential to a perfect differential in that one variable. 

122. As illustrations, take (x, y) as the co-ordinates 
of a point P in the plane XOY, then the path of inte- 
gration is shown by a path of P from its initial to its 
final position. In this case the imperfect differential 
y dx is the differential area described by the ordinate y, 

and I y dx along a given path is the total, continuously 

described, ordinate area of the path. For, P, P' being 



112 PRIMER OF CALCULUS 

two successive positions (pc,y) (x',y f ) of the point on the 
path between its initial and final positions, then the 
ordinate area PLL' P f =y 1 Ax, where y 1 is some ordi- 
nate between y= LP and y'=UP\ is the typical 
difference f(x, Az, y, Ay) whose sum is the total ordinate 
area. Thus ^ v b y 1 Ax denotes the total ordinate area, 
described by any n successive changes along the given 
path. As we take n larger and larger, such area is 
described more and more nearly continuously, i.e., 
by sums of smaller and smaller differences, so that 

y dx represents the result of con- 
tinuous summation of ordinate areas described by con- 
tinuous motion along the path. If along the given 

path y=cfi'x, then the integral area becomes j <£' xdx 

which can be determined when a function <f>x can 

be found such that d<j>x = <)>'xdx viz., it will be 

In 
cjix=cfiu — <f>a. The path must be given in order to 



evaluate j y dx. Another imperfect differential is the 

differential of radial area, ^(xdy — ydx); the integral of 
this between given limits is the total area described by 
the radius OP from the initial to the final position of P, 
and it requires the path to be known, such as by u y a 
given function of x" or u x, y given functions of 0," etc., 
before the integral is determinate. On the contrary, 
the sum of the ordinate and radial area described by 
OPL is independent of the path, and is %(uv — ab), 



INTEGRATION 113 

since it is the integral, J I (xdy-\-ydx)=% j d(xy), the 
integral of a perfect differential. 

123. As a physical illustration, the amount of heat 
required to expand a gas from volume x and pressure y 
to volume x' and pressure ?/' depends upon the path or 
series of continuous changes from the condition (x, y) 
to the condition (x',y'). The differential amount of 
heat is the amount that would be required to change 
from the condition (x, y) to the condition (x-\-dx, y-\-dy) 
if changes continued as at (x, y), and is a quantity 
Ldx-\-Mdy, where L, M are functions of (x,y); when 
dx, dy are very small this is the principal part of any of 
the actual amounts of heat required to make the change. 
Because the amount of heat absorbed varies with the 
path, this differential cannot be a perfect differential. 

124. Potential. On the contrary the work done by 
a given natural field of force in displacing a given 
particle along any path is independent of the path 
when the terminal and initial points are the same. If 
X, Y, Z be the components of the force in the directions 
of the axes of x, y, z acting on the particle in the posi- 
tion P= (x, y, 2), then, the differential work for any dis- 
placement ds = sum of the differential work of each 
component = Xdx-\-Ydy-\-Zdz, which must be a 
perfect differ en tial. 

It thus appears that X, Y, Z must be the partial deri- 
vatives of some function <j> (x, y, z) as to x, y, z, and 
— cf> (Xj y, z) is called the potential of the field on the 
particle P. The existence of such a function was dis- 



114 PRIMER OF CALCULUS 

covered by Lagrange ; several years later Greene pointed 
out that it represented potential energy, or energy of 
position in the field with reference to the initial posi- 
tion; viz., — cf>(x,y,z)=c is the equation of the surface 
of potential c, and c is the amount of work that will be 
done by the field in moving the particle along any path 
from this surface to the zero potential surface, viz., 

J div==jd<j>(x,y,z) = \<j>(x,y,z)=c. 

Examples, 

1. Find the sums from x=o to cc— 4 of a: 2 Ax, 
(x 2 -\-Ax) Ax, A.£ 3 /3, for uniform variations of 2, 4, 
and n changes. Also represent in each case the terms 
of the sum by rectangles in the plane XOY, of ordinates 
y = x 2 , y=x 2 -{-Ax, y=A(^x z )/Ax respectively, and 
base Ax. (Representation by ordinate areas). 

2. Show that the limit of each sum in Ex. 1 for 
n = co is 64/3; also verify that the proportionals of 
the differenences approach the same differential x 2 dx. 
Show that in the ordinate area representation, the 
common limit of these sums is the ordinate area of the 
parabola y=x 2 from x=o to x=4. 

3. Find lim %^xAx by uniform variation; also geomet- 
rically by its ordinate area representation; also by the 
fundamental theorem of integration (Art. 119). 

4. Find by direct integration the functions whose 
differentials are udu, u 2 du, and that vanish when u 
vanishes. 

Ans. u 2 /2, u*/3. (See Art. 114). 



INTEGRATION 115 

u 2 — a 2 



xdx 
also by Th. 3 and d. x*/2=xdx. 

6. Verify by ordinate area that J *J(a 2 — x 2 )dx= 
-a 2 /A; also by Th. 3 and Ex. 14 p. 43. 

7. Verify by ordinate area that I J(a 2 — x 2 )dx = 

iw^a^-tt^ + ^sin- 1 -, = d~K J(a 2 —u 2 )du. 

A G, 

8. If the ordinates of two curves to any same abscissa 
are in a constant ratio b/a, then their ordinate areas 
between the same bounding ordinates are in the ratio 
b/a. 

9. If parallel chords between two curves vary as their 
distances from a fixed point, then the area of a segment 
between two chords as bases is that of the rectangle on 
the altitude and the half sum of the bases. 

[A chord at distance x is ex, and the area between 

chords at distances x=a, x=u is j cxdx = 
(u — a) (cu-\-cd)/2.~\ 

10. If the areas of parallel sections of a tubular surface 
vary as the squares of their distances from a fixed point, 
find the volume between two parallel sections in terms 
of the bases b 1 , 6 3 , middle section 6 2 , and altitude h. 

Ans. A(6 1 +46 2 +6 3 )/6. 

11. Find the volume of a cone or pyramid in terms 
of its base and altitude, and also the distance of its 
center of volume from the base. 

Ans. bh/3, h/i. 



116 PRIMER OF CALCULUS 

12. Find the volume of a hemisphere of radius a, 
and the distance of its center of volume from its base. 

2- a* 



■j>< 



•)dX: 



3 ' 

3a 



morri V/ V= -r— - I £(a 2 — a; 2 )da;= 
2a 3 J v y 



13. Find the moment of inertia of a right circular 
cylinder of altitude c and radius a, about its axis. 



f. 



x 2 . 2xcxdx= — — 

2 



14. A wedge is made from a right circular cylinder 
of radius a and altitude h, by plane sections through a 
diameter of one base and the tangents to the other base 
that are parallel to such diameter; find the volume of 
the wedge. [A plane perpendicular to the diameter at 
distance x from the axis cuts each side piece that is 
taken off to make the wedge in a triangle whose area is 
to ah/ 2 as a 2 — x 2 is to a 2 , by similar triangles; 

thus V=7ra 2 h— 4 C ~{a 2 — x 2 ) &=a 2 A (tt— f).] 

15. Find the volume common to two circular cylin- 
ders of common upper base and tangent lower bases. 

2 C - (a 2 —x 2 ) dx=± a 2 h. 

16. The axes of two right circular cylinders intersect 
at right angles; find the included volume, and the 
surface. 

17. A sphere of radius a is charged with A.na 2 l units 
of electricity (^ per unit area); find its potential at a 
point C whose distance is c from the center. 



INTEGRATION 117 

Draw the diameter OC, and let CP=r, <OCP=0 
where P is any point of a small circle of the sphere 
abont OC as axis. The charge of the zone between the 
circle P = (r, 0) and the circle P'=r' ) 0' (whose altitude 
is A. r cos 0) is 2-aAA.rcos# and its potential at C 
is 2ttgU A (rcos 0)/i\ where r 1 is an average disiance 
between r and r'. Since r 2 -\-c 2 — 2r c cos 0= a 2 , there- 
fore rdr=cd. r cos 0, and the differential potential is 
2r aid (r cos 0) /r = 2~ al dr/c. When c > a then r 
changes from c — a to c-\-a giving the total potential 
A.~a 2 l/c. When c<a then r changes from a — c to 
a-\-c giving the total potential Aizal. 

18. Simpson's Rule. When given the end chords 
y lt ?/ 8 , the middle chord y 2 , and the distance h be- 
tween chords, of an area between parallel chords, the 
generally best approximate value of the area from these 
data is h (y l -f 4y 2 +7/ 3 )/3. 

Any chord y is a function of x, its distance from the 
middle chord, say y = a -\- bx -\- ex 2 +..., a convergent 
series. The three given chords can determine at most 
only three co-efficients, i.e., the best approximation is 
to determine y to three terms, and neglect the others 
as probably very small. From y=y 1 , y 2J y zi when 
x = — h, o, h, we have y x =a — bh-\-ch 2 , y 2 =a, 
y s =a-\-bh-\-ch 2 , and consequently y 1 -\-y z =2a-\-2ch 2 . 
The required area is 

When given 5 equi-distant chords y lt y 2i y 8 , y A , y 5i 



118 PRIMER OF CALCULUS 

then from the first and last three, the area is by this 
rule h(y x + 4y a +2y 8 -f-4y 4 +y 6 )/3, and so on. 
This rule applies when the y's are the areas of equi- 
distant parallel sections of a volume, etc. 

10. If 2/ 1 , y 2i 2/ 3 , y 4 , are four parallel chords of an area 
at equal distances 2h, then 3A(y 1 + 3y 2 -+3y. 8 +y 4 )/'§ 
is generally the best approximation to the area from 
chord y 1 to chord y 4 . 

[If y be the chord at distance x from the central 
chord of the area, then the best assumption is 
y = a -f- bx -f- ex 2 -|- ex 3 , etc.] 

20. C^sinx 2n dx= fccosx* n dx 



f 2 sin x 2n dx= C 



_ 1. 3. 5... (2 n— 1) 7T 
— 2. 4. 6... 2^ ' 2 

/ T sin £ 2 " +1 dx— § "2 cos x 2w+1 da; 
J 

2.4.6... 2rc 

= 3.5.7...2n + l [63c ' e] 

21. If ??i ? n be positive integers and p=iz/a then 
I cosm^sc. cos npx. dx = o or -Ja, according as m, n 

are unequal or equal. 

22. If fx = ^A + A r cos px+A 2 cos2px+ A 3 cos3px-\-... 
for all values of x from to a=7r/p 5 then show that 

2 /- a 



^4 n =- j fx cos npx. dx. 
a J 



[This is Fourier's theorem for expansion of any 
function in a series of cosines.] 



SUCCESSIVE DIFFERENTIATION 119 



CHAPTER V 
Successive Differentiation 

125. Any differential quantity may be itself differ- 
entiated, and the result again differentiated, and so on, 
since any differential is a new variable depending upon 
the previous independent variables and their differen- 
tials. Thus each differentiation introduces as many 
new independent variables as there are original or 
primary independent variables. 

E. g., differentiating (according to Prin. 3) first 
partially as to x and then partially as to y and adding, 
we find 

d . xy = d x . y -\- xdy 
d 2 . xy = d 2 x . y -\-dx . dy 

-\-dx . dy -\-xd 2 y 
= d 2 x . y -f 2 dx . dy + x . d 2 y 
d z .xy = d 3 x. y -{-2 d 2 x . dy-\-dx.d 2 y 

+ d 2 x .dy + 2 dx . d 2 y + xdHj 
= d*x.y-{-3d 2 x.dy-{-3 dx . d 2 y + xd*y 

The identity of these successive operations, so far as 
numerical co-efficients are concerned, with successive 
multiplications of a -j- b by a-\-b, shows that we must 
have always binomial co-efficients, or that 

(a) d n . xy= d n x . y-\-nd n - 1 x. dy+ n (n—1) d n ~ 2 x.d 2 y-\-... 
-\-xd n y. This analogy results from d n = (d x -\-d y ) n , 
and the fact (Prin. 4 following.) that d x , d y obey the 
same formal laws of combination as numbers. 



120 PRIMER OF CALCULUS 

126. The successive differentiation d n = (lim NA) n 
can be defined as a single process lim N n A n . To 
explain A n , let x take any variation of n changes from 
its original value, say x, x 15 x 2 , ... x ft , This determines 
a variation of n — 1 changes in Ax, a variation of n — 2 
changes in A 2 x, and so on to one value of A n x. If E 
symbolize the process of changing from one value of a 
variation to the next (enlarging the variable), so that 
Ex = x^ E 2 x = Ex 1 =x 2 , E s x = x 3 , ... E n x = x n , 
then we shall always have A= E — 1, (understanding 
that a sum of processes, means the summing of results 
of each process) i. e. y 

Ax = (E— 1) x = Ex — 1 . x = Xj _x 

A 2 x=: (E—l) (x l — x) =x 2 — x 1 — x 1 -\-x = x 2 — 2x x + x 

A 3 x= (x s — 2x 2 -\-x 1 ) — (x 2 +2x 2 — x) = x 3 — 3x 2 — 3x x — x 

and 

(n— 1) 

A 7l x=(£'— l) ra x=x n — nx ?l _i-f-n — ^ — -% n — 2 -*-nx 1 ±x 

A 

Note. — It is because A, E, 1 are processes that are 
distributive over sums and commutative with each 
other, that their laws of combination are like those of 
ordinary numbers. It is different with processes "log" 
and ll J" that do not obey these laws. E. g., 
(log + V) 2 * = (log + V) (log x + Jx) = log (log x + Jx) 
+ J (log x + Jx) , but not =log log x + 2 log ^/x-f- «/ ^/x, 
since it is not true that log (x + y) — log # + log y, 
J(x + y) = Jx + 7y and log ^x = ^ log x. 

Conversely, the w values Ax, A 2 x, ... A w x determine 
the variation of x, viz., 



SUCCESSl YE DIFFERENTIA TION 121 

x n — E n x=(l + &y n x = x + n&x-\- n ^ n 7~ ' A 2 x-\- ... 
+ ??A ?i - 1 x -\- A n x. 

127. In the case of an independent variable x, the n 

successive differences Ax, A 2 x, ... A^x.are assignable 
at will, and can each be made to approach zero in such a 
way that for any proportional factor N that approaches 
infinity, the proportional differences NAx, N 2 A 2 x, 
... N n A w x shall approach any assigned limits dx, d 2 x, 
... d n x. At the same time, the successive proportional 
differences NAw, N 2 A 2 iu ... N n A n w, of a successively 
differentiable dependent variable iv, must approach 
limits dtv, d 2 w, ... d n iv, that depend only upon the 
values of the independent variables and their successive 
differentials. 

E.g., Wxy = x 2 y 2 — 2x 1 y 1 +xy 

= (x+2Ax+A 2 x) ( 2/+ 2A?/+A, 2 2/)-2(a:+Ax) (y+Ay)+xy 

= x A 2 y -j- 2 AxAy + yA 2 x + 2 Ax A 2 ?/ + 2 A 2 x Ay 4. A 2 xA 2 y 

so that d 2 . xy = lim iV 2 A 2 .xy — xd 2 y -\-2dxdy -\-yd 2 x. 

Partial Differentiation 

128. Differentiation under the suppositions that 
certain variable quantities are constants is called partial 
differentiation. When the suppositions affect only in- 
dependent variables and not all of those, then partial 
differentiation of equals give equals. Thus in 
(x -\- y) 2 = x 2 -f- 2ry -\- y 2 , we may consider either x or 
y to vary alone, or both to vary together so- that xy is 
constant, and the differentials of each member are 
equal under any of these suppositions. But in 



122 PRIMER OF CALCULUS 

x 2 -\-y 2 =4:, where both variables must change in 
order to maintain the equality, partial differentiation 
as to x or y does not give an equation that follows from 
the given one. 

Prin. 4. Two successive partial differentiations of any 
function are commutative in order of operation. 

In symbols d t d 2 w = d 2 d x iv, where d x affects certain 
variables x, ... of w, and d 2 affects certain variables 
y, ... of w. 

By Prin. 3 d 2 w = (d y + . . .) w =d y w -f- . . . 

and d 1 d 2 iv = (d x + . . . ) d 2 iv = d x d y w + . . . 
Similajly d 1 w = (d x -{-...) iv = d x iv -\- ... 

and d 2 d 1 w= (d y -{-...) d 1 w= d y d x iv + ... 

It therefore only remains to prove the principle for 
partial differentiations as to any two variables x, y, and 
this is done at once by 

d x d y f(x, y) = d x lim N[f(x, y') —f(x, yj\ (definition) 
= lim N [d x f(x, y') — d x f(x, ?/)] (Prin 2) 
= d y d x f(x, y) (definition) 

By dividing this result by dx dy, we find, 

9 div 9 9w 

' 9x 9y 9y 9x 

129. It is not possible to mark a differential symbol 
so as to show all possiole suppositions under which the 
differentiation is taken, and form is often used instead 
of marks for the purpose. It follows that changes of 
form that are legitimate when the differentials are suf- 
ficiently marked to indicate their significance, are not 



PARTIAL DIFFERENTIATIOX 123 

legitimate when the form is changed so as to lose its 
assigned significance as a mark of a certain kind of 
differentiation. For example, the early practice was, 
with a function w of x, y, to use div/dx, dw/dy as 
forms for partial differentiation as to x, y, so that in 

this notation dw = ~-j- dx -f- ~r~^V' To cancel dx, dy 

here gives the incorrect result dw = 2dw; but if we 
mark the differentiations by subscripts, then dx, dy, 
may be cancelled, giving dw = d z w-\-d y w, a correct 
result. 

130. Again, if y be a function of x, whose successive 
derivatives are y', y" , ... so that dy = y'dx, dy' = y f, dx,... 
then d 2 y = y" dx* + y' d 2 x, 

d*y = y'" dx* + Zy" dx d 2 x + y' d*x, etc. 
If we suppose dx to be constant in successive differen- 
tiations then d 2 y = y" dx 2 , d s y = y'"dx*,... and the 
latter forms are understood as indicating partial differen- 
tiatiation with dx assumed constant; so that the quotients 
d 2 y/dx 2 =y", d z y / dx z =y f,r become abbreviated 
forms for the successive derivatives of y as to x. The 
full forms for these derivitives are 
d dy d d dy 
dx dx } dx dx dx' 

in which d is unrestricted. Since every differentiation, 
in these successive derivations, is performed upon a 
function of x alone, it follows that it is independent of 
differentials and may be performed under the supposi- 
tion that any differential we please is constant; in 
particular dx = constant reduces them by Prin. 2 to 



124 PRIMER OF CALCULUS 

the above abbreviated forms. If x is not the inde- 
pendent variable but is a function of the independent 
variable z, say x =fz, then we may transform deriva- 
tives as to x into derivatives as to z by substituting 
dx =f'z dz in the complete forms, but not in the abbrevi- 

atedones. Thus, -7- -M- =--—- -rr-^-, but d ! y = y" dx 2 
' dxdx f'zdz f'zdz' * * 

does not become d 2 y = y"f'z 2 dz 2 because the accepted 

significance of the latter form is that dz == constant 

whereas for the first, it is that dx = constant, so that 

the two symbols d 2 y are not symbols of the same 

quantity. The true equality resulting from dx— f'zdz 

is here, (d 2 y) dx constant = y' f'z 2 dz 2 . 

, TT , , d 2 y d dy d 2 y dx — d 2 xdy. 

We have always -=-£■ = ~r- ~r — — r~o m 

dx 2 ax dx ax* 

which, as pointed out above, the differentiations of the 
last form are unrestricted, and may be made as to any 
differential a constant. In particular to make y the in- 
dependent variable, we let dy = constant, and so find 

d 2 y _ dy* d 2 x 
dx 2 ~~ dx*' dy 2 ' 

Examples IV 

1. y = x 2 e x ; d 2 y/dx 2 =e x (x 2 -f-4z + 4) 

2. y = x 5 ; d 5 y/dx 5 = 5\ 

3. y = x z \ogx-, d 4 y/dxA—5/x 

4. y = log sin x; d 2 y/dx 2 = — csc£ 2 

5. y = e~ mx (a cos nx + b sin nx) ; 

dx 2 ^ dx~ y ' J * 



PARTIAL DIFFERENTIA TION 125 

6. y = er mx (a + bx); 

d 2 y , rt dy , 

7. y = ae- mx + be~ nx ; 

d 2 y . ■ , ^dy , 

d 2 v dy 

8. Solve the equation -=-|- -\- A -j- -\- By = o where 

A , i? are constants. 

Try y = e cx , whence c 2 -\- Ac -\- B = o to determine c. 
Show that the sum of two solutions, each multiplied 
by an arbitrary constant, is a solution, and thence, if 
c= — m, — n, derive the solution of Ex. 7. This solu- 
tion is general, because it involves two arbitrary con- 
stants a, b, such as would be obtained by two successive 
anti-differentiations. If m = n show that besides e~ mx 
also xe~ mx is a solution, so that the general solution is 
that of Ex. 5. Ifc = — m-\-nJ — 1, — m — nj — 1, 
then replace the Iwo solutions e cx by their sum multi- 
plied by 1/2 and their difference multiplied by 
1/2 J — 1, and so obtain the general solution of Ex. 5. 

d 2 y dy 

9. Iiy==f 1 x be a solution of -^- -\- A -~-\-By=fx 

find the general solution. 

[Let y=f 1 x-\-z be the general solution; whence z is 

found from -^-- - 4- A -j- -4- B z = o.~\ 
ax 2 ■ ax J 

10. Solve ^ -|-2 ^ + 26 y = 154 cos 4z + 8 sin 4x. 

Ans. y = 9 cos 4x -f- 8 sin 4x -)- e~ x (a cos 5a: -f- b sin 5x). 



126 PRIMER OF CALCULUS 

11 ax. ^ \ dn dHu . d^udv . 

11. Showthat^^ == ^-,+n^ =i ^ + . 

d d ) n 



. dfx ■ dx j 

12. Show that- T ^e a */a=e a *(a + - T -) w /':r. 

cfa ?l J K ] d,x J ' 

[Verify for n==l and thence for n = 2 } 3, ...] 

d n 

13. -= — e ax x 2 =e ax \a n x 2 4-2n a™- 1 x-\-n (n— l)a n - 2 ~\ 

dx 11 J 

14. Verify the following transformations of inde- 
pendent variable, and solve the equations. 

d 2 y cfy 2 ~ efr/ 3 d 2 £ . cfo . 

d 2 y 2x d\i , v , d 2 v 

d^ + 1+^ Tx +o+ ^ = ° t0 a? +? / = ' *= tan 2 - 

Successive Integration 

131. We consider certain "multiple" integrals in 
two or more variables, where the integration is partial 
with respect to each variable in turn, and as if the 
remaining variables were constants, while the limits of 
variation of each variable are constants, or at most 
functions of the variables that are assumed as con- 
stants. The integral is written so that the order of 
successive integration is from right to left, so that each 
integration reduces the multiple integral to one of next 
lower multiplicity. Thus 

Jj(x, y) dxdy=J a | J J(x, y) dy J dx. 



SUCCESSIVE INTEGRATION 127 

is a double integral. The bracketed integral is first 
evaluated, with x as a constant, and b, v are at most 
functions of x, so that the bracketed integral is simply a 
4>x term in a single integral. Similarly 

a J J e F(?,V,')dxdydz 

« J 6 { J c ^ (X ' V ' ^ ^ j ^^ 

is a triple integral. The bracketed integral is first 
evaluated, with x, y as constants, and c, w are at most 
functions of x, y, so that the bracketed integral is 
simply an f(x, y) term in a double integral. Similarly 
for multiple integrals in four or more variables. 

132. The single, double, and triple integrals have 
geometric representation as the limits of sums of dif- 
ferences over a line, surface, and volume respectively. 
We illustrate by examples worked out in detail, show- 
ing also in the first illustration the sums considered, of 
which the integrals are limits for continuous variation. 
The student should make the drawings as described. 

133. Hind the moment of inertia of a rectangular paral- 
lelopiped of edges a, 6, c, about the edge c as axis. 

Let OA = a, OB = b, be the two edges in the plane 
of the diagram. Confine attention to the rectangle AB, 
knowing that a length c of the volume is above every 
point P. Take OA, OB as axes of x, y. Lay off 
OL = x, OU = x\ LP=y, L'P'=y', where P, P' 
are neighboring points within the rectangle AB. Then 
cAxky is the difference volume whose base is the 



128 PRIMER OF CALCULUS 

rectangle PP' and (x 2 -|- y 2 ) cAr Ay is its moment of 
inertia as to the axis OC, where (x 1? y x ) is some point 
of the rectangle PP'. Here, x 2 -\-y 2 is a function, 
<£ (x, Ax, y, Ay), that reduces to x 2 -\-y 2 when Ax = o, 
Ay = o. If we assume x, Ax to be constants and give 
y any variation from o to 6, then the sum of the differ- 
ence volumes PP', is S cAx Ay, = 6cAx, the difference 
volume whose base is the rectangle LU Q' Q of altitude 
b and base Ax; and the sum of the moments of inertia 
of the difference volumes PP' is S (x 2 -|-y 2 ) rAxAy= 
the moment of inertia of the difference volume LQ'. 
Next give x any variation from o to a, and then the 
sum of the difference volume LQ', is ^ be A x = a b c = 
the entire volume; and the sum of their moments 
of inertia is the entire moment of inertia, 
% a Sj (x 2 + y 2 )cA xA y. The results hold for any vari- 
ation, first of y from o to b with x, Ax constant, and 
then of x from o to a; and in particular for continuous 
variations, in which the sums become integrals of 
the differentials corresponding to the vanishing differ- 
ences. Thus cdxdy is the differential volume P, and 
(x 2 -\-y 2 ) cdx dy is its moment of inertia. Also, 
cdxdy = bcdx is the differential valume LQ, while 

Jb b 2 

(x 2 -f-y 2 ) cdx dy = bc (x 2 + -o~) dx is its moment 

of inertia; and finally, j ( cdxdy = 1 bcdx=abc ) 



SUCCESSIVE INTEGRATION 129 

is the total volume, and) j (x 2 -\-y 2 )c dxdy = 

«y o J o 

J a b 2 a 2 -\-b 2 
be (x 2 +-o~ ) dx = abc ^- — is the total 

moment of inertia. 

134. Find the moment of inertia of a right cylinder about 
its axis, OC = c, ivhere the base is an elliptic quadrant of 
radii OA = a, OB = b. 

As before cdxdy, c (x 2 -\-y 2 ) dxdy are the differ- 
ential volume at P and its moment of inertia; and these 
integrals trom y = o to y = y 1 = LK= bj(a 2 — x 2 ) / a 
= the ordinate of the point K on the ellipse AB whose 
abscissa is x — 0L, are 



/: 



/: 



l c dx dy = cy x dx, 





\ (x 2 +7/ 2 ) dxdy = c (x 2 y x +%-) dx. 



3 

These are therefore respectively the differential vol- 
ume LK and its moment of inertia. Finally 

-abc 



mom. mer 

- abc a 2 + b 2 



volume = J cy x dx = — j— [Ex. 14, p. 43.] 

•= PcO^i+y) dx 

[Ex. 16, p. 43.] 

135. Find the moment of inertia of the preceding elliptic 
cylinder about the axis OA. 

Show the axis OC and the cylinder COAB in pers- 
pective on the plane of the diagram. Draw a plane 



130 PRIMER OF CALCULUS 

through the ordinate LK perpendicular to OA, and 
therefore intersecting the cylinder in elements KN, LM; 
draw RQ parallel to LM and intersecting LK in R, 
MN in Q; and on RQ take any point P=x,y,z = 
(OL, LR, RP). The differential volume P is dx dy dz, 
and its moment of inertia as to OA is (y 2 -{- z 2 ) dx dy dz. 
Integrating from 2 = to z = c we find the differential 
volume RQ and its moment of inertia^ 

dx dy dz = c dx dy, 



1 



r. 



c 6 

o G/ 2 -f 2 2 ) dx dy dz = (y 2 c+ y) dx dy. 



Integrating again from 

?/— to y —y 1 =LK=b *J a 2 — x 2 /a 

we find the differential volume KLMN and its moment 
of inertia, 

cy 1 dx, cy 1 ^-^—dx 

These are results that can be found directly from 
geometry and Art. 134. Integrating then from x = o 
to x = a, we find the total volume and its moment of 

inertia, 

irrabc, i7rabc(b 2 /4: + c 2 /3.) 

136. Find the volume and moment of inertia about 00 
of the octant of an ellipsoid of radii OA === a, OB = 6, 
OC=c. 

The figure is shown in plane diagram by projections 
of quadrants of the ellipses AB, BO, OA; the section by 
a plane perpendicular to OA through the ordinate LK of 
arc AB is an elliptic quadrant KM from AB to OA; 



SUCCESSIVE INTEGRATION 131 

draw in this plane RQ parallel to LM from LK to arc 

x 2 y 2 2 2 
JO/, and we have since — + 77 + -y = 1 for any point 

(x } y, z) on the ellipsoid, LK— b J (a 2 — x 2 )/a, 

x 2 2 2 
RQ = c,J ( 1 — — — v— ). Take on i? Q any point 

P= ( r? y f z) = ( 0L, LR, RP). The volumes and 
moments of inertia about OC are, 

•a /'LA' /\RQ 



I dx c/t/ cfe 

«7 



and the same integration of (V 2 -f- 2/ 2 ) dxdy dz. 

The results of the first two partial integrations may be 
found geometrically from Art. 134, applied to the 
elliptic cylinder of length dx on the base LKM, and are 

- LJf . LM (fa, T LJT . LIT dx. -V or [since 

4 4 4 L 

LM = c J ( a 2 — x 2 ) / a], ^— ^ ( a 2 — x 2 ) dx, 

-^ — J- — - (a 2 — x 2 ) 2 dx, and the integrals of these 

j. . n ,, Ttabc nabc (a 2 -\-b 2 ) 

irom x— to x=a give finallv —* — , -77— =? -• 

bo 5 

137. If m denote the volume (or mass of a homo- 
geneous volume) then the rectangular parallelopiped, 
elliptic cylinder, and ellipsoid, whose semi-axes of 
symmetry are OA = a, OB = b, OC=c, and of which 
the volumes of the preceding articles are octants, have 
the following moments of inertia 



132 PRIMER OF CALCULUS - 

Parallelopiped : m — ^- — about OC ; etc. 

a 2 -j-6 2 
Cylinder : m — -J- — about OC ; 

b 2 c 2 
m (— -f- -^) about (L4, etc. 

Ellipsoid : m 5 — about 00 ; etc. 

These are easily remembered, and are useful, 
especially in connection with the theorem of Art. 139. 

138. Center of gravity. The differential element of 
mass at (x, y, z) called a particle P, is u dx dy dz where 
u is in general a function of x, y, z denoting the density 
at that point. If (x, y, z) be the center of mass (center 
of gravity) then computing moments as to the planes 
yz, zx xy directly, and by the sum of the moments of the 
particles, we find 

mx= I I I uxdxdydz 



my= I I I uy dx dy dz 
mz= I I | uz dx dy dz 



139. The moment of inertia of a given mass about a 
given axis is equal to the moment of inertia of the mass about 
a paralled axes through the center of mass plies the product 
of the mass and the square of the distance oetween the axes. 

Let OZ be the axis through the center of mass, and 
let OA = a, be the distance between the axes. We 

have I f I ux dx dy dz = m x = o by hypothesis ; and 



S UCCESSIVE INTEGRA TION 133 

therefore m k 2 = I ( J [ (x — a) 2 -f- y 2 ~\ u cfo cfa/ tfz = 

f f f C^ 2 + 2/ 2 ) w ^ x cfy ck + a 2 ( ( f u dx dy dz = 

m &* -|-ma 2 . 

MO. A cylinder with elements perpendicular to the 
x, y plane intersects a surface z =/ (x, y) ; required the 
surface area intercepted by the cylinder. 

If A 2 S be the portion of the surface intercepted by 
the cylinder on the rectangle Ax Ay as base, the cor- 
responding differential element of surface d 2 S is inter- 
cepted on the tangent plane to the surface at P=(x, y,z) 
by the cylinder on dx dy as base; so that if y be the 
angle between this tangent plane and the plane x y, 
i. e., the angle between the axis of z and the normal 
to the surface at P, we have d 2 Scosy = dxdy so 

that d 2 S=dxdy sec y, and S= f I sec ydxdy, over 

the base of the given cylinder on the x y plane. To 
find sec y in terms of x, y observe that if iv = o be the 
equation of a surface, then for variations of P on that 

surface w = o or -=— dx -4- -=— dy -I — =- dz = o. This 
dx x dy dz 

shows that the line PN whose components on the axes 

are TV - ? -?r-, ~*r is perpendicular to the tangent line 

PS = ds, whose components are dx, dy, dz.* 

Thus PN is a normal to the surface at P since it 

* If r, r' be two lines that make an angle A with each other and angles 
a,b,c, a',b',c', with the axes, then eqating the projection of r upon r' 
to the sum of the projections of the components of r upon r\ we have 

r cos A = r cos a . cos a' + r cos 5 . cos b' + r cos c . cos c' 
or rr / cos A = I I' -r mm' + n n' in terms of the components of r, r' ; if 
this is zero then A is a right angle. 



134 PRIMER OF CALCULUS 

is perpendicular to every tangent line PS, and 

PN cos y = 9w/9z,—l when w = z — f(x,y). Also 

$ w 2 Q w 2 Q w I 2 .% 12 3 2 2 

PN 2 = ^- 4-^ + °^- =l+t U?- where 



2 , 9w 


2 , 5m; 


2 , . ^2 | 2 , 9Z 


+¥ 


+ 37 


= 1 + — - 4- — — 

, ' 9x 1 cty 



5 a; 
z — /(*, 30- 

EXAMPLES. 

1. Find the center of gravity of an arc of a circle of 
radius a and length £. 

[Take the center as origin and axis of y parallel to 
the chord, whose length is &=2asin -1 (l/2a). Then 

k 



xds\ I ds= I ady 
since y varies from — Jfc to \k when s varies from o to J. 



fe* 



yds\ I ds= I — a 
o ' •/ o J h 



dx 



l=o. 



2. Find the center of gravity of a straight wire whose 
density varies uniformly from end to end. 

Let a, b, be the densities at the ends, and I be the length 
of the wire; then the density at distance x from the 
first end is u=a-\-(b — a)x/l; and 

X= I uxdx\ | udx=^l(a-\-2b)/(a-\-b) 

•SO | *J 

3. Find the center of gravity of the first quadrant arc 
of the hypocycloid x*^\-y%=a\. 

X=y=fa. 

4. Find the center of gravity of the first quadrant 
area of the circle x 2 -\-y 2 =a 2 . 



SUCCESSIVE INTEGRATION 135 

I xdxdy\ II cfad?/ 

o J o \ *J oy o 

4a 

5. Find the center of gravity of the first quadrant 
area of the hypocycloid x%-\-yi=a%. 

6. Find the center of gravity of the area of the 
cardioid r = 2a(l — cos0). 

7. Find the center of gravity of the parahloic area 
y 2 =4ax from x=o. 

8. Find the center of gravity of a hemisphere whose 
density varies as the distance from the center; find also 
its radius of gyration about its axis of symmetry. 

9. Find the center of gravity and the radius of 
gyration of the volume generated by the revolution of 
y 2 =4ax about the axis of x (from x=o). 

10. Find the radius of gyration of a sphere about a 
tangent line as axis. (Arts. 137, 239). 

11. Compare and verify when necessary the following 
formulas for straight and rotary motion of a rigid body; 
dm = mass of a differential particle; v = its linear 
velocity; r = its distance from the axis in rotary 
motion. The integration extends to every particle of 
the body, and for this integration v = constant in linear 
motion, and v/r=w = constant in rotary motion; v 
and w are, however, variable with the time. 



136 



A PRIMER OF CALCULUS 



STRAIGHT MOTION 


ROTARY MOTION 


m= inertia (mass) 


I=moment of inertia 


= 1 dm 


= j r 2 dm 


v= velocity 


w=v/r= angular vel. 


mv= momentum 


Iw=mo. of momentum 


= I vdm 


= I rvdm 


-—= acceleration 
dt 


-j— = angular acceleration 


dv r 
m -r = force 
dt 


I -7-= moment of forces 
dt 


=Si dm 


=$ r 'i dm 


±mv 2 — kinetic energy 


^Iw 2 = kinetic energy 


= I \v 2 dm 


= f %v 2 dm 



In finding the moment of the forces in rotary motion 

dv 
the tangential acceleration of dm, viz, -^-, is the only 

effective component, since the normal component v 2 /r 
passes through the axis. (See Art. 86). In straight 
motion there is no normal component of acceleration 
since the curvature of a straight line is zero (or its radius 
of curvature infinite). 

12. Find the surfaces cut from one another by a right 
circular cylinder of radius a and a sphere of radius 2a, 
whose center is on the cylindrical surface. 

13. Find the volume enclosed in Ex. 12. 



SUCCESSIVE INTEGRATION 137 

14. Find the volume enclosed by the surface 

x%-\-y%-\-zi==a* 

15. Show that if an arc of length s be revolved about 
an axis in its plane, the surface described is the product 
of s into the length of path of the center of gravity of s. 
State and prove a similar theorem for revolution of a 
plane area about an axis in its plane. 

16. Find the volume and surface of the anchor ring, 
generated by revolving a circle of radius 6, about an 
axis in its plane at distance a from the center. 

17. In polar co-ordinates (r, 0, <£) of a point P with 
reference to an initial line OA and initial plane OAB, 
we have 

r=OP, = <AOP, <f> = 

diedral angle between OAP and OAB. Show that the 
differential element of volume is 

d V= dr .rdd.r sin 6 d <£. 
==r 2 sin ddrdGd<l>. 



138 A PRIMER OF CALCULUS 

Rules of Differentiation 

I. d.u l =lu l — 1 du. 

(a) d.u l v m =u l — 1 v m - 1 (lvdu-{-mudv). 

(b) d.u 1 /v m =u l ~ 1 Q J vdu — mudv)/v m + 1 . 

II. d. av =av log ady; d.ey=eydy. 

(a) d . uy =yuy~ 1 du -f- vP log w . dy. 

III. cnog a v=cfa;/'?;loga; d\ogv=dv/v. 

(a) dlog (z-f Vx 2 + ra 2 ) =da;/ V(^ 2 + ca% )- 



(b) c?log 



+ Ja 2 +cx 2 ^(as+cz*)' 



, s ,, a-\-x 2adx 

(c) dlog — — = -. 

w ° a—-x a 2 — x 2 

IV dsmv = co8V.dv; dcosv= — sinv.dv. 

dta,nv=secv 2 dv; dcotv= — cscv 2 dv. 

dsecv=secv. tsmv. dv; dcscv= — esc v. cot v. dv. 

V dsm- 1 -=-7- — =— dcos- 1 -. 

a J(a 2 — x 2 ) a 

c?tan —1 - = 



d SeC -1 - = tt-a jr = — d CSC -1 -. 

a x,J(x 2 — a 2 ) a 

,x dx , . . x — a 

CtVerS -1 ~= -ry^ — = d Sin" 1 

a J(2ax — x 2 ) a 

Prin 3. df{x, y) = d x f(x, y)+d y f(x, y). 




RULES OF INTEGRATION 



139 



Rules of Integration 

I d- 1 .u l du=u l+1 /(l+l). 

Ill d~ 1 .du/u=logu, or log — u, or log c it. 

111 ( a )/ V (J+ca-y) =1°« (* + V*~ C ^), etc. 

/( a 2_ a; 2)— S111 a* 



/ 



dx 



J(2ax— 

/dx 
a 2 + X 2 



■x 2 ) 



vers -1 -, or sin" 
a 



a; — a 



1, t x 

-tan -1 -. 
a a 



ttt r \ C & x 1 i a + * 

in (b)J- d * 



/» dr 



1, 

- log - 



\/.r 2 -\-cx' 



xj{& 



1 1 x 

a 2 ) x a 



II 
IV 
III 

III 

IV 

IV 



d~ 1 . a x dx = a x /log a ; d~ 1 . e* c/x = e x . 
d -1 . sin vdv = — cos v ; d— 1 . cos v dv = sin v. 
d~ 1 . tan v . dv = — log cos v ; 
d~ 1 . cot vdv = log sin v. 
cZ — 1 . sec vdv = log (sec -y + tan 0) ; 
d _1 . esc vdv = log (esc — cot v) . 
d~ x sec v 2 c?y = tan t? ; dh^csc v 2 cfo = — cot v. 
d~ 1 sec v tan v dv = sec i>; 
d —1 esc v cot v c?v = — esc v. 
d~ x . sin v 2 dv= \ (2v — sin 2v) ; 
c/ —1 cos v 2 dv = J- (2v ' + sin 2i?). 
Inv. Prin. 3. d-+. d x f(x,y)=f(x,y)—d^d y f(x,y). 



NOV 29 1902 



